NumberDB$^\beta$ Advanced search Tags Tables Help Login
Bernstein basis polynomials $b_{v,n}$
edit · history · discussion · files · long url · analysis polynomial
Numbers
$n$
$v$ 
0
0:
1
1
0:
-x + 1
1
1:
x
2
0:
x^2 - 2*x + 1
2
1:
-2*x^2 + 2*x
2
2:
x^2
3
0:
-x^3 + 3*x^2 - 3*x + 1
3
1:
3*x^3 - 6*x^2 + 3*x
3
2:
-3*x^3 + 3*x^2
3
3:
x^3
4
0:
x^4 - 4*x^3 + 6*x^2 - 4*x + 1
4
1:
-4*x^4 + 12*x^3 - 12*x^2 + 4*x
4
2:
6*x^4 - 12*x^3 + 6*x^2
4
3:
-4*x^4 + 4*x^3
4
4:
x^4
5
0:
-x^5 + 5*x^4 - 10*x^3 + 10*x^2 - 5*x + 1
5
1:
5*x^5 - 20*x^4 + 30*x^3 - 20*x^2 + 5*x
5
2:
-10*x^5 + 30*x^4 - 30*x^3 + 10*x^2
5
3:
10*x^5 - 20*x^4 + 10*x^3
5
4:
-5*x^5 + 5*x^4
5
5:
x^5
6
0:
x^6 - 6*x^5 + 15*x^4 - 20*x^3 + 15*x^2 - 6*x + 1
6
1:
-6*x^6 + 30*x^5 - 60*x^4 + 60*x^3 - 30*x^2 + 6*x
6
2:
15*x^6 - 60*x^5 + 90*x^4 - 60*x^3 + 15*x^2
6
3:
-20*x^6 + 60*x^5 - 60*x^4 + 20*x^3
6
4:
15*x^6 - 30*x^5 + 15*x^4
6
5:
-6*x^6 + 6*x^5
6
6:
x^6
7
0:
-x^7 + 7*x^6 - 21*x^5 + 35*x^4 - 35*x^3 + 21*x^2 - 7*x + 1
7
1:
7*x^7 - 42*x^6 + 105*x^5 - 140*x^4 + 105*x^3 - 42*x^2 + 7*x
7
2:
-21*x^7 + 105*x^6 - 210*x^5 + 210*x^4 - 105*x^3 + 21*x^2
7
3:
35*x^7 - 140*x^6 + 210*x^5 - 140*x^4 + 35*x^3
7
4:
-35*x^7 + 105*x^6 - 105*x^5 + 35*x^4
7
5:
21*x^7 - 42*x^6 + 21*x^5
7
6:
-7*x^7 + 7*x^6
7
7:
x^7
8
0:
x^8 - 8*x^7 + 28*x^6 - 56*x^5 + 70*x^4 - 56*x^3 + 28*x^2 - 8*x + 1
8
1:
-8*x^8 + 56*x^7 - 168*x^6 + 280*x^5 - 280*x^4 + 168*x^3 - 56*x^2 + 8*x
8
2:
28*x^8 - 168*x^7 + 420*x^6 - 560*x^5 + 420*x^4 - 168*x^3 + 28*x^2
8
3:
-56*x^8 + 280*x^7 - 560*x^6 + 560*x^5 - 280*x^4 + 56*x^3
8
4:
70*x^8 - 280*x^7 + 420*x^6 - 280*x^5 + 70*x^4
8
5:
-56*x^8 + 168*x^7 - 168*x^6 + 56*x^5
8
6:
28*x^8 - 56*x^7 + 28*x^6
8
7:
-8*x^8 + 8*x^7
8
8:
x^8
9
0:
-x^9 + 9*x^8 - 36*x^7 + 84*x^6 - 126*x^5 + 126*x^4 - 84*x^3 + 36*x^2 - 9*x + 1
9
1:
9*x^9 - 72*x^8 + 252*x^7 - 504*x^6 + 630*x^5 - 504*x^4 + 252*x^3 - 72*x^2 + 9*x
9
2:
-36*x^9 + 252*x^8 - 756*x^7 + 1260*x^6 - 1260*x^5 + 756*x^4 - 252*x^3 + 36*x^2
9
3:
84*x^9 - 504*x^8 + 1260*x^7 - 1680*x^6 + 1260*x^5 - 504*x^4 + 84*x^3
9
4:
-126*x^9 + 630*x^8 - 1260*x^7 + 1260*x^6 - 630*x^5 + 126*x^4
9
5:
126*x^9 - 504*x^8 + 756*x^7 - 504*x^6 + 126*x^5
9
6:
-84*x^9 + 252*x^8 - 252*x^7 + 84*x^6
9
7:
36*x^9 - 72*x^8 + 36*x^7
9
8:
-9*x^9 + 9*x^8
9
9:
x^9
10
0:
x^10 - 10*x^9 + 45*x^8 - 120*x^7 + 210*x^6 - 252*x^5 + 210*x^4 - 120*x^3 + 45*x^2 - 10*x + 1
10
1:
-10*x^10 + 90*x^9 - 360*x^8 + 840*x^7 - 1260*x^6 + 1260*x^5 - 840*x^4 + 360*x^3 - 90*x^2 + 10*x
10
2:
45*x^10 - 360*x^9 + 1260*x^8 - 2520*x^7 + 3150*x^6 - 2520*x^5 + 1260*x^4 - 360*x^3 + 45*x^2
10
3:
-120*x^10 + 840*x^9 - 2520*x^8 + 4200*x^7 - 4200*x^6 + 2520*x^5 - 840*x^4 + 120*x^3
10
4:
210*x^10 - 1260*x^9 + 3150*x^8 - 4200*x^7 + 3150*x^6 - 1260*x^5 + 210*x^4
10
5:
-252*x^10 + 1260*x^9 - 2520*x^8 + 2520*x^7 - 1260*x^6 + 252*x^5
10
6:
210*x^10 - 840*x^9 + 1260*x^8 - 840*x^7 + 210*x^6
10
7:
-120*x^10 + 360*x^9 - 360*x^8 + 120*x^7
10
8:
45*x^10 - 90*x^9 + 45*x^8
10
9:
-10*x^10 + 10*x^9
10
10:
x^10
11
0:
-x^11 + 11*x^10 - 55*x^9 + 165*x^8 - 330*x^7 + 462*x^6 - 462*x^5 + 330*x^4 - 165*x^3 + 55*x^2 - 11*x + 1
11
1:
11*x^11 - 110*x^10 + 495*x^9 - 1320*x^8 + 2310*x^7 - 2772*x^6 + 2310*x^5 - 1320*x^4 + 495*x^3 - 110*x^2 + 11*x
11
2:
-55*x^11 + 495*x^10 - 1980*x^9 + 4620*x^8 - 6930*x^7 + 6930*x^6 - 4620*x^5 + 1980*x^4 - 495*x^3 + 55*x^2
11
3:
165*x^11 - 1320*x^10 + 4620*x^9 - 9240*x^8 + 11550*x^7 - 9240*x^6 + 4620*x^5 - 1320*x^4 + 165*x^3
11
4:
-330*x^11 + 2310*x^10 - 6930*x^9 + 11550*x^8 - 11550*x^7 + 6930*x^6 - 2310*x^5 + 330*x^4
11
5:
462*x^11 - 2772*x^10 + 6930*x^9 - 9240*x^8 + 6930*x^7 - 2772*x^6 + 462*x^5
11
6:
-462*x^11 + 2310*x^10 - 4620*x^9 + 4620*x^8 - 2310*x^7 + 462*x^6
11
7:
330*x^11 - 1320*x^10 + 1980*x^9 - 1320*x^8 + 330*x^7
11
8:
-165*x^11 + 495*x^10 - 495*x^9 + 165*x^8
11
9:
55*x^11 - 110*x^10 + 55*x^9
11
10:
-11*x^11 + 11*x^10
11
11:
x^11
12
0:
x^12 - 12*x^11 + 66*x^10 - 220*x^9 + 495*x^8 - 792*x^7 + 924*x^6 - 792*x^5 + 495*x^4 - 220*x^3 + 66*x^2 - 12*x + 1
12
1:
-12*x^12 + 132*x^11 - 660*x^10 + 1980*x^9 - 3960*x^8 + 5544*x^7 - 5544*x^6 + 3960*x^5 - 1980*x^4 + 660*x^3 - 132*x^2 + 12*x
12
2:
66*x^12 - 660*x^11 + 2970*x^10 - 7920*x^9 + 13860*x^8 - 16632*x^7 + 13860*x^6 - 7920*x^5 + 2970*x^4 - 660*x^3 + 66*x^2
12
3:
-220*x^12 + 1980*x^11 - 7920*x^10 + 18480*x^9 - 27720*x^8 + 27720*x^7 - 18480*x^6 + 7920*x^5 - 1980*x^4 + 220*x^3
12
4:
495*x^12 - 3960*x^11 + 13860*x^10 - 27720*x^9 + 34650*x^8 - 27720*x^7 + 13860*x^6 - 3960*x^5 + 495*x^4
12
5:
-792*x^12 + 5544*x^11 - 16632*x^10 + 27720*x^9 - 27720*x^8 + 16632*x^7 - 5544*x^6 + 792*x^5
12
6:
924*x^12 - 5544*x^11 + 13860*x^10 - 18480*x^9 + 13860*x^8 - 5544*x^7 + 924*x^6
12
7:
-792*x^12 + 3960*x^11 - 7920*x^10 + 7920*x^9 - 3960*x^8 + 792*x^7
12
8:
495*x^12 - 1980*x^11 + 2970*x^10 - 1980*x^9 + 495*x^8
12
9:
-220*x^12 + 660*x^11 - 660*x^10 + 220*x^9
12
10:
66*x^12 - 132*x^11 + 66*x^10
12
11:
-12*x^12 + 12*x^11
12
12:
x^12
13
0:
-x^13 + 13*x^12 - 78*x^11 + 286*x^10 - 715*x^9 + 1287*x^8 - 1716*x^7 + 1716*x^6 - 1287*x^5 + 715*x^4 - 286*x^3 + 78*x^2 - 13*x + 1
13
1:
13*x^13 - 156*x^12 + 858*x^11 - 2860*x^10 + 6435*x^9 - 10296*x^8 + 12012*x^7 - 10296*x^6 + 6435*x^5 - 2860*x^4 + 858*x^3 - 156*x^2 + 13*x
13
2:
-78*x^13 + 858*x^12 - 4290*x^11 + 12870*x^10 - 25740*x^9 + 36036*x^8 - 36036*x^7 + 25740*x^6 - 12870*x^5 + 4290*x^4 - 858*x^3 + 78*x^2
13
3:
286*x^13 - 2860*x^12 + 12870*x^11 - 34320*x^10 + 60060*x^9 - 72072*x^8 + 60060*x^7 - 34320*x^6 + 12870*x^5 - 2860*x^4 + 286*x^3
13
4:
-715*x^13 + 6435*x^12 - 25740*x^11 + 60060*x^10 - 90090*x^9 + 90090*x^8 - 60060*x^7 + 25740*x^6 - 6435*x^5 + 715*x^4
13
5:
1287*x^13 - 10296*x^12 + 36036*x^11 - 72072*x^10 + 90090*x^9 - 72072*x^8 + 36036*x^7 - 10296*x^6 + 1287*x^5
13
6:
-1716*x^13 + 12012*x^12 - 36036*x^11 + 60060*x^10 - 60060*x^9 + 36036*x^8 - 12012*x^7 + 1716*x^6
13
7:
1716*x^13 - 10296*x^12 + 25740*x^11 - 34320*x^10 + 25740*x^9 - 10296*x^8 + 1716*x^7
13
8:
-1287*x^13 + 6435*x^12 - 12870*x^11 + 12870*x^10 - 6435*x^9 + 1287*x^8
13
9:
715*x^13 - 2860*x^12 + 4290*x^11 - 2860*x^10 + 715*x^9
13
10:
-286*x^13 + 858*x^12 - 858*x^11 + 286*x^10
13
11:
78*x^13 - 156*x^12 + 78*x^11
13
12:
-13*x^13 + 13*x^12
13
13:
x^13
14
0:
x^14 - 14*x^13 + 91*x^12 - 364*x^11 + 1001*x^10 - 2002*x^9 + 3003*x^8 - 3432*x^7 + 3003*x^6 - 2002*x^5 + 1001*x^4 - 364*x^3 + 91*x^2 - 14*x + 1
14
1:
-14*x^14 + 182*x^13 - 1092*x^12 + 4004*x^11 - 10010*x^10 + 18018*x^9 - 24024*x^8 + 24024*x^7 - 18018*x^6 + 10010*x^5 - 4004*x^4 + 1092*x^3 - 182*x^2 + 14*x
14
2:
91*x^14 - 1092*x^13 + 6006*x^12 - 20020*x^11 + 45045*x^10 - 72072*x^9 + 84084*x^8 - 72072*x^7 + 45045*x^6 - 20020*x^5 + 6006*x^4 - 1092*x^3 + 91*x^2
14
3:
-364*x^14 + 4004*x^13 - 20020*x^12 + 60060*x^11 - 120120*x^10 + 168168*x^9 - 168168*x^8 + 120120*x^7 - 60060*x^6 + 20020*x^5 - 4004*x^4 + 364*x^3
14
4:
1001*x^14 - 10010*x^13 + 45045*x^12 - 120120*x^11 + 210210*x^10 - 252252*x^9 + 210210*x^8 - 120120*x^7 + 45045*x^6 - 10010*x^5 + 1001*x^4
14
5:
-2002*x^14 + 18018*x^13 - 72072*x^12 + 168168*x^11 - 252252*x^10 + 252252*x^9 - 168168*x^8 + 72072*x^7 - 18018*x^6 + 2002*x^5
14
6:
3003*x^14 - 24024*x^13 + 84084*x^12 - 168168*x^11 + 210210*x^10 - 168168*x^9 + 84084*x^8 - 24024*x^7 + 3003*x^6
14
7:
-3432*x^14 + 24024*x^13 - 72072*x^12 + 120120*x^11 - 120120*x^10 + 72072*x^9 - 24024*x^8 + 3432*x^7
14
8:
3003*x^14 - 18018*x^13 + 45045*x^12 - 60060*x^11 + 45045*x^10 - 18018*x^9 + 3003*x^8
14
9:
-2002*x^14 + 10010*x^13 - 20020*x^12 + 20020*x^11 - 10010*x^10 + 2002*x^9
14
10:
1001*x^14 - 4004*x^13 + 6006*x^12 - 4004*x^11 + 1001*x^10
14
11:
-364*x^14 + 1092*x^13 - 1092*x^12 + 364*x^11
14
12:
91*x^14 - 182*x^13 + 91*x^12
14
13:
-14*x^14 + 14*x^13
14
14:
x^14
15
0:
-x^15 + 15*x^14 - 105*x^13 + 455*x^12 - 1365*x^11 + 3003*x^10 - 5005*x^9 + 6435*x^8 - 6435*x^7 + 5005*x^6 - 3003*x^5 + 1365*x^4 - 455*x^3 + 105*x^2 - 15*x + 1
15
1:
15*x^15 - 210*x^14 + 1365*x^13 - 5460*x^12 + 15015*x^11 - 30030*x^10 + 45045*x^9 - 51480*x^8 + 45045*x^7 - 30030*x^6 + 15015*x^5 - 5460*x^4 + 1365*x^3 - 210*x^2 + 15*x
15
2:
-105*x^15 + 1365*x^14 - 8190*x^13 + 30030*x^12 - 75075*x^11 + 135135*x^10 - 180180*x^9 + 180180*x^8 - 135135*x^7 + 75075*x^6 - 30030*x^5 + 8190*x^4 - 1365*x^3 + 105*x^2
15
3:
455*x^15 - 5460*x^14 + 30030*x^13 - 100100*x^12 + 225225*x^11 - 360360*x^10 + 420420*x^9 - 360360*x^8 + 225225*x^7 - 100100*x^6 + 30030*x^5 - 5460*x^4 + 455*x^3
15
4:
-1365*x^15 + 15015*x^14 - 75075*x^13 + 225225*x^12 - 450450*x^11 + 630630*x^10 - 630630*x^9 + 450450*x^8 - 225225*x^7 + 75075*x^6 - 15015*x^5 + 1365*x^4
15
5:
3003*x^15 - 30030*x^14 + 135135*x^13 - 360360*x^12 + 630630*x^11 - 756756*x^10 + 630630*x^9 - 360360*x^8 + 135135*x^7 - 30030*x^6 + 3003*x^5
15
6:
-5005*x^15 + 45045*x^14 - 180180*x^13 + 420420*x^12 - 630630*x^11 + 630630*x^10 - 420420*x^9 + 180180*x^8 - 45045*x^7 + 5005*x^6
15
7:
6435*x^15 - 51480*x^14 + 180180*x^13 - 360360*x^12 + 450450*x^11 - 360360*x^10 + 180180*x^9 - 51480*x^8 + 6435*x^7
15
8:
-6435*x^15 + 45045*x^14 - 135135*x^13 + 225225*x^12 - 225225*x^11 + 135135*x^10 - 45045*x^9 + 6435*x^8
15
9:
5005*x^15 - 30030*x^14 + 75075*x^13 - 100100*x^12 + 75075*x^11 - 30030*x^10 + 5005*x^9
15
10:
-3003*x^15 + 15015*x^14 - 30030*x^13 + 30030*x^12 - 15015*x^11 + 3003*x^10
15
11:
1365*x^15 - 5460*x^14 + 8190*x^13 - 5460*x^12 + 1365*x^11
15
12:
-455*x^15 + 1365*x^14 - 1365*x^13 + 455*x^12
15
13:
105*x^15 - 210*x^14 + 105*x^13
15
14:
-15*x^15 + 15*x^14
15
15:
x^15
16
0:
x^16 - 16*x^15 + 120*x^14 - 560*x^13 + 1820*x^12 - 4368*x^11 + 8008*x^10 - 11440*x^9 + 12870*x^8 - 11440*x^7 + 8008*x^6 - 4368*x^5 + 1820*x^4 - 560*x^3 + 120*x^2 - 16*x + 1
16
1:
-16*x^16 + 240*x^15 - 1680*x^14 + 7280*x^13 - 21840*x^12 + 48048*x^11 - 80080*x^10 + 102960*x^9 - 102960*x^8 + 80080*x^7 - 48048*x^6 + 21840*x^5 - 7280*x^4 + 1680*x^3 - 240*x^2 + 16*x
16
2:
120*x^16 - 1680*x^15 + 10920*x^14 - 43680*x^13 + 120120*x^12 - 240240*x^11 + 360360*x^10 - 411840*x^9 + 360360*x^8 - 240240*x^7 + 120120*x^6 - 43680*x^5 + 10920*x^4 - 1680*x^3 + 120*x^2
16
3:
-560*x^16 + 7280*x^15 - 43680*x^14 + 160160*x^13 - 400400*x^12 + 720720*x^11 - 960960*x^10 + 960960*x^9 - 720720*x^8 + 400400*x^7 - 160160*x^6 + 43680*x^5 - 7280*x^4 + 560*x^3
16
4:
1820*x^16 - 21840*x^15 + 120120*x^14 - 400400*x^13 + 900900*x^12 - 1441440*x^11 + 1681680*x^10 - 1441440*x^9 + 900900*x^8 - 400400*x^7 + 120120*x^6 - 21840*x^5 + 1820*x^4
16
5:
-4368*x^16 + 48048*x^15 - 240240*x^14 + 720720*x^13 - 1441440*x^12 + 2018016*x^11 - 2018016*x^10 + 1441440*x^9 - 720720*x^8 + 240240*x^7 - 48048*x^6 + 4368*x^5
16
6:
8008*x^16 - 80080*x^15 + 360360*x^14 - 960960*x^13 + 1681680*x^12 - 2018016*x^11 + 1681680*x^10 - 960960*x^9 + 360360*x^8 - 80080*x^7 + 8008*x^6
16
7:
-11440*x^16 + 102960*x^15 - 411840*x^14 + 960960*x^13 - 1441440*x^12 + 1441440*x^11 - 960960*x^10 + 411840*x^9 - 102960*x^8 + 11440*x^7
16
8:
12870*x^16 - 102960*x^15 + 360360*x^14 - 720720*x^13 + 900900*x^12 - 720720*x^11 + 360360*x^10 - 102960*x^9 + 12870*x^8
16
9:
-11440*x^16 + 80080*x^15 - 240240*x^14 + 400400*x^13 - 400400*x^12 + 240240*x^11 - 80080*x^10 + 11440*x^9
16
10:
8008*x^16 - 48048*x^15 + 120120*x^14 - 160160*x^13 + 120120*x^12 - 48048*x^11 + 8008*x^10
16
11:
-4368*x^16 + 21840*x^15 - 43680*x^14 + 43680*x^13 - 21840*x^12 + 4368*x^11
16
12:
1820*x^16 - 7280*x^15 + 10920*x^14 - 7280*x^13 + 1820*x^12
16
13:
-560*x^16 + 1680*x^15 - 1680*x^14 + 560*x^13
16
14:
120*x^16 - 240*x^15 + 120*x^14
16
15:
-16*x^16 + 16*x^15
16
16:
x^16
17
0:
-x^17 + 17*x^16 - 136*x^15 + 680*x^14 - 2380*x^13 + 6188*x^12 - 12376*x^11 + 19448*x^10 - 24310*x^9 + 24310*x^8 - 19448*x^7 + 12376*x^6 - 6188*x^5 + 2380*x^4 - 680*x^3 + 136*x^2 - 17*x + 1
17
1:
17*x^17 - 272*x^16 + 2040*x^15 - 9520*x^14 + 30940*x^13 - 74256*x^12 + 136136*x^11 - 194480*x^10 + 218790*x^9 - 194480*x^8 + 136136*x^7 - 74256*x^6 + 30940*x^5 - 9520*x^4 + 2040*x^3 - 272*x^2 + 17*x
17
2:
-136*x^17 + 2040*x^16 - 14280*x^15 + 61880*x^14 - 185640*x^13 + 408408*x^12 - 680680*x^11 + 875160*x^10 - 875160*x^9 + 680680*x^8 - 408408*x^7 + 185640*x^6 - 61880*x^5 + 14280*x^4 - 2040*x^3 + 136*x^2
17
3:
680*x^17 - 9520*x^16 + 61880*x^15 - 247520*x^14 + 680680*x^13 - 1361360*x^12 + 2042040*x^11 - 2333760*x^10 + 2042040*x^9 - 1361360*x^8 + 680680*x^7 - 247520*x^6 + 61880*x^5 - 9520*x^4 + 680*x^3
17
4:
-2380*x^17 + 30940*x^16 - 185640*x^15 + 680680*x^14 - 1701700*x^13 + 3063060*x^12 - 4084080*x^11 + 4084080*x^10 - 3063060*x^9 + 1701700*x^8 - 680680*x^7 + 185640*x^6 - 30940*x^5 + 2380*x^4
17
5:
6188*x^17 - 74256*x^16 + 408408*x^15 - 1361360*x^14 + 3063060*x^13 - 4900896*x^12 + 5717712*x^11 - 4900896*x^10 + 3063060*x^9 - 1361360*x^8 + 408408*x^7 - 74256*x^6 + 6188*x^5
17
6:
-12376*x^17 + 136136*x^16 - 680680*x^15 + 2042040*x^14 - 4084080*x^13 + 5717712*x^12 - 5717712*x^11 + 4084080*x^10 - 2042040*x^9 + 680680*x^8 - 136136*x^7 + 12376*x^6
17
7:
19448*x^17 - 194480*x^16 + 875160*x^15 - 2333760*x^14 + 4084080*x^13 - 4900896*x^12 + 4084080*x^11 - 2333760*x^10 + 875160*x^9 - 194480*x^8 + 19448*x^7
17
8:
-24310*x^17 + 218790*x^16 - 875160*x^15 + 2042040*x^14 - 3063060*x^13 + 3063060*x^12 - 2042040*x^11 + 875160*x^10 - 218790*x^9 + 24310*x^8
17
9:
24310*x^17 - 194480*x^16 + 680680*x^15 - 1361360*x^14 + 1701700*x^13 - 1361360*x^12 + 680680*x^11 - 194480*x^10 + 24310*x^9
17
10:
-19448*x^17 + 136136*x^16 - 408408*x^15 + 680680*x^14 - 680680*x^13 + 408408*x^12 - 136136*x^11 + 19448*x^10
17
11:
12376*x^17 - 74256*x^16 + 185640*x^15 - 247520*x^14 + 185640*x^13 - 74256*x^12 + 12376*x^11
17
12:
-6188*x^17 + 30940*x^16 - 61880*x^15 + 61880*x^14 - 30940*x^13 + 6188*x^12
17
13:
2380*x^17 - 9520*x^16 + 14280*x^15 - 9520*x^14 + 2380*x^13
17
14:
-680*x^17 + 2040*x^16 - 2040*x^15 + 680*x^14
17
15:
136*x^17 - 272*x^16 + 136*x^15
17
16:
-17*x^17 + 17*x^16
17
17:
x^17
18
0:
x^18 - 18*x^17 + 153*x^16 - 816*x^15 + 3060*x^14 - 8568*x^13 + 18564*x^12 - 31824*x^11 + 43758*x^10 - 48620*x^9 + 43758*x^8 - 31824*x^7 + 18564*x^6 - 8568*x^5 + 3060*x^4 - 816*x^3 + 153*x^2 - 18*x + 1
18
1:
-18*x^18 + 306*x^17 - 2448*x^16 + 12240*x^15 - 42840*x^14 + 111384*x^13 - 222768*x^12 + 350064*x^11 - 437580*x^10 + 437580*x^9 - 350064*x^8 + 222768*x^7 - 111384*x^6 + 42840*x^5 - 12240*x^4 + 2448*x^3 - 306*x^2 + 18*x
18
2:
153*x^18 - 2448*x^17 + 18360*x^16 - 85680*x^15 + 278460*x^14 - 668304*x^13 + 1225224*x^12 - 1750320*x^11 + 1969110*x^10 - 1750320*x^9 + 1225224*x^8 - 668304*x^7 + 278460*x^6 - 85680*x^5 + 18360*x^4 - 2448*x^3 + 153*x^2
18
3:
-816*x^18 + 12240*x^17 - 85680*x^16 + 371280*x^15 - 1113840*x^14 + 2450448*x^13 - 4084080*x^12 + 5250960*x^11 - 5250960*x^10 + 4084080*x^9 - 2450448*x^8 + 1113840*x^7 - 371280*x^6 + 85680*x^5 - 12240*x^4 + 816*x^3
18
4:
3060*x^18 - 42840*x^17 + 278460*x^16 - 1113840*x^15 + 3063060*x^14 - 6126120*x^13 + 9189180*x^12 - 10501920*x^11 + 9189180*x^10 - 6126120*x^9 + 3063060*x^8 - 1113840*x^7 + 278460*x^6 - 42840*x^5 + 3060*x^4
18
5:
-8568*x^18 + 111384*x^17 - 668304*x^16 + 2450448*x^15 - 6126120*x^14 + 11027016*x^13 - 14702688*x^12 + 14702688*x^11 - 11027016*x^10 + 6126120*x^9 - 2450448*x^8 + 668304*x^7 - 111384*x^6 + 8568*x^5
18
6:
18564*x^18 - 222768*x^17 + 1225224*x^16 - 4084080*x^15 + 9189180*x^14 - 14702688*x^13 + 17153136*x^12 - 14702688*x^11 + 9189180*x^10 - 4084080*x^9 + 1225224*x^8 - 222768*x^7 + 18564*x^6
18
7:
-31824*x^18 + 350064*x^17 - 1750320*x^16 + 5250960*x^15 - 10501920*x^14 + 14702688*x^13 - 14702688*x^12 + 10501920*x^11 - 5250960*x^10 + 1750320*x^9 - 350064*x^8 + 31824*x^7
18
8:
43758*x^18 - 437580*x^17 + 1969110*x^16 - 5250960*x^15 + 9189180*x^14 - 11027016*x^13 + 9189180*x^12 - 5250960*x^11 + 1969110*x^10 - 437580*x^9 + 43758*x^8
18
9:
-48620*x^18 + 437580*x^17 - 1750320*x^16 + 4084080*x^15 - 6126120*x^14 + 6126120*x^13 - 4084080*x^12 + 1750320*x^11 - 437580*x^10 + 48620*x^9
18
10:
43758*x^18 - 350064*x^17 + 1225224*x^16 - 2450448*x^15 + 3063060*x^14 - 2450448*x^13 + 1225224*x^12 - 350064*x^11 + 43758*x^10
18
11:
-31824*x^18 + 222768*x^17 - 668304*x^16 + 1113840*x^15 - 1113840*x^14 + 668304*x^13 - 222768*x^12 + 31824*x^11
18
12:
18564*x^18 - 111384*x^17 + 278460*x^16 - 371280*x^15 + 278460*x^14 - 111384*x^13 + 18564*x^12
18
13:
-8568*x^18 + 42840*x^17 - 85680*x^16 + 85680*x^15 - 42840*x^14 + 8568*x^13
18
14:
3060*x^18 - 12240*x^17 + 18360*x^16 - 12240*x^15 + 3060*x^14
18
15:
-816*x^18 + 2448*x^17 - 2448*x^16 + 816*x^15
18
16:
153*x^18 - 306*x^17 + 153*x^16
18
17:
-18*x^18 + 18*x^17
18
18:
x^18
19
0:
-x^19 + 19*x^18 - 171*x^17 + 969*x^16 - 3876*x^15 + 11628*x^14 - 27132*x^13 + 50388*x^12 - 75582*x^11 + 92378*x^10 - 92378*x^9 + 75582*x^8 - 50388*x^7 + 27132*x^6 - 11628*x^5 + 3876*x^4 - 969*x^3 + 171*x^2 - 19*x + 1
19
1:
19*x^19 - 342*x^18 + 2907*x^17 - 15504*x^16 + 58140*x^15 - 162792*x^14 + 352716*x^13 - 604656*x^12 + 831402*x^11 - 923780*x^10 + 831402*x^9 - 604656*x^8 + 352716*x^7 - 162792*x^6 + 58140*x^5 - 15504*x^4 + 2907*x^3 - 342*x^2 + 19*x
19
2:
-171*x^19 + 2907*x^18 - 23256*x^17 + 116280*x^16 - 406980*x^15 + 1058148*x^14 - 2116296*x^13 + 3325608*x^12 - 4157010*x^11 + 4157010*x^10 - 3325608*x^9 + 2116296*x^8 - 1058148*x^7 + 406980*x^6 - 116280*x^5 + 23256*x^4 - 2907*x^3 + 171*x^2
19
3:
969*x^19 - 15504*x^18 + 116280*x^17 - 542640*x^16 + 1763580*x^15 - 4232592*x^14 + 7759752*x^13 - 11085360*x^12 + 12471030*x^11 - 11085360*x^10 + 7759752*x^9 - 4232592*x^8 + 1763580*x^7 - 542640*x^6 + 116280*x^5 - 15504*x^4 + 969*x^3
19
4:
-3876*x^19 + 58140*x^18 - 406980*x^17 + 1763580*x^16 - 5290740*x^15 + 11639628*x^14 - 19399380*x^13 + 24942060*x^12 - 24942060*x^11 + 19399380*x^10 - 11639628*x^9 + 5290740*x^8 - 1763580*x^7 + 406980*x^6 - 58140*x^5 + 3876*x^4
19
5:
11628*x^19 - 162792*x^18 + 1058148*x^17 - 4232592*x^16 + 11639628*x^15 - 23279256*x^14 + 34918884*x^13 - 39907296*x^12 + 34918884*x^11 - 23279256*x^10 + 11639628*x^9 - 4232592*x^8 + 1058148*x^7 - 162792*x^6 + 11628*x^5
19
6:
-27132*x^19 + 352716*x^18 - 2116296*x^17 + 7759752*x^16 - 19399380*x^15 + 34918884*x^14 - 46558512*x^13 + 46558512*x^12 - 34918884*x^11 + 19399380*x^10 - 7759752*x^9 + 2116296*x^8 - 352716*x^7 + 27132*x^6
19
7:
50388*x^19 - 604656*x^18 + 3325608*x^17 - 11085360*x^16 + 24942060*x^15 - 39907296*x^14 + 46558512*x^13 - 39907296*x^12 + 24942060*x^11 - 11085360*x^10 + 3325608*x^9 - 604656*x^8 + 50388*x^7
19
8:
-75582*x^19 + 831402*x^18 - 4157010*x^17 + 12471030*x^16 - 24942060*x^15 + 34918884*x^14 - 34918884*x^13 + 24942060*x^12 - 12471030*x^11 + 4157010*x^10 - 831402*x^9 + 75582*x^8
19
9:
92378*x^19 - 923780*x^18 + 4157010*x^17 - 11085360*x^16 + 19399380*x^15 - 23279256*x^14 + 19399380*x^13 - 11085360*x^12 + 4157010*x^11 - 923780*x^10 + 92378*x^9
19
10:
-92378*x^19 + 831402*x^18 - 3325608*x^17 + 7759752*x^16 - 11639628*x^15 + 11639628*x^14 - 7759752*x^13 + 3325608*x^12 - 831402*x^11 + 92378*x^10
19
11:
75582*x^19 - 604656*x^18 + 2116296*x^17 - 4232592*x^16 + 5290740*x^15 - 4232592*x^14 + 2116296*x^13 - 604656*x^12 + 75582*x^11
19
12:
-50388*x^19 + 352716*x^18 - 1058148*x^17 + 1763580*x^16 - 1763580*x^15 + 1058148*x^14 - 352716*x^13 + 50388*x^12
19
13:
27132*x^19 - 162792*x^18 + 406980*x^17 - 542640*x^16 + 406980*x^15 - 162792*x^14 + 27132*x^13
19
14:
-11628*x^19 + 58140*x^18 - 116280*x^17 + 116280*x^16 - 58140*x^15 + 11628*x^14
19
15:
3876*x^19 - 15504*x^18 + 23256*x^17 - 15504*x^16 + 3876*x^15
19
16:
-969*x^19 + 2907*x^18 - 2907*x^17 + 969*x^16
19
17:
171*x^19 - 342*x^18 + 171*x^17
19
18:
-19*x^19 + 19*x^18
19
19:
x^19
20
0:
x^20 - 20*x^19 + 190*x^18 - 1140*x^17 + 4845*x^16 - 15504*x^15 + 38760*x^14 - 77520*x^13 + 125970*x^12 - 167960*x^11 + 184756*x^10 - 167960*x^9 + 125970*x^8 - 77520*x^7 + 38760*x^6 - 15504*x^5 + 4845*x^4 - 1140*x^3 + 190*x^2 - 20*x + 1
20
1:
-20*x^20 + 380*x^19 - 3420*x^18 + 19380*x^17 - 77520*x^16 + 232560*x^15 - 542640*x^14 + 1007760*x^13 - 1511640*x^12 + 1847560*x^11 - 1847560*x^10 + 1511640*x^9 - 1007760*x^8 + 542640*x^7 - 232560*x^6 + 77520*x^5 - 19380*x^4 + 3420*x^3 - 380*x^2 + 20*x
20
2:
190*x^20 - 3420*x^19 + 29070*x^18 - 155040*x^17 + 581400*x^16 - 1627920*x^15 + 3527160*x^14 - 6046560*x^13 + 8314020*x^12 - 9237800*x^11 + 8314020*x^10 - 6046560*x^9 + 3527160*x^8 - 1627920*x^7 + 581400*x^6 - 155040*x^5 + 29070*x^4 - 3420*x^3 + 190*x^2
20
3:
-1140*x^20 + 19380*x^19 - 155040*x^18 + 775200*x^17 - 2713200*x^16 + 7054320*x^15 - 14108640*x^14 + 22170720*x^13 - 27713400*x^12 + 27713400*x^11 - 22170720*x^10 + 14108640*x^9 - 7054320*x^8 + 2713200*x^7 - 775200*x^6 + 155040*x^5 - 19380*x^4 + 1140*x^3
20
4:
4845*x^20 - 77520*x^19 + 581400*x^18 - 2713200*x^17 + 8817900*x^16 - 21162960*x^15 + 38798760*x^14 - 55426800*x^13 + 62355150*x^12 - 55426800*x^11 + 38798760*x^10 - 21162960*x^9 + 8817900*x^8 - 2713200*x^7 + 581400*x^6 - 77520*x^5 + 4845*x^4
20
5:
-15504*x^20 + 232560*x^19 - 1627920*x^18 + 7054320*x^17 - 21162960*x^16 + 46558512*x^15 - 77597520*x^14 + 99768240*x^13 - 99768240*x^12 + 77597520*x^11 - 46558512*x^10 + 21162960*x^9 - 7054320*x^8 + 1627920*x^7 - 232560*x^6 + 15504*x^5
20
6:
38760*x^20 - 542640*x^19 + 3527160*x^18 - 14108640*x^17 + 38798760*x^16 - 77597520*x^15 + 116396280*x^14 - 133024320*x^13 + 116396280*x^12 - 77597520*x^11 + 38798760*x^10 - 14108640*x^9 + 3527160*x^8 - 542640*x^7 + 38760*x^6
20
7:
-77520*x^20 + 1007760*x^19 - 6046560*x^18 + 22170720*x^17 - 55426800*x^16 + 99768240*x^15 - 133024320*x^14 + 133024320*x^13 - 99768240*x^12 + 55426800*x^11 - 22170720*x^10 + 6046560*x^9 - 1007760*x^8 + 77520*x^7
20
8:
125970*x^20 - 1511640*x^19 + 8314020*x^18 - 27713400*x^17 + 62355150*x^16 - 99768240*x^15 + 116396280*x^14 - 99768240*x^13 + 62355150*x^12 - 27713400*x^11 + 8314020*x^10 - 1511640*x^9 + 125970*x^8
20
9:
-167960*x^20 + 1847560*x^19 - 9237800*x^18 + 27713400*x^17 - 55426800*x^16 + 77597520*x^15 - 77597520*x^14 + 55426800*x^13 - 27713400*x^12 + 9237800*x^11 - 1847560*x^10 + 167960*x^9
20
10:
184756*x^20 - 1847560*x^19 + 8314020*x^18 - 22170720*x^17 + 38798760*x^16 - 46558512*x^15 + 38798760*x^14 - 22170720*x^13 + 8314020*x^12 - 1847560*x^11 + 184756*x^10
20
11:
-167960*x^20 + 1511640*x^19 - 6046560*x^18 + 14108640*x^17 - 21162960*x^16 + 21162960*x^15 - 14108640*x^14 + 6046560*x^13 - 1511640*x^12 + 167960*x^11
20
12:
125970*x^20 - 1007760*x^19 + 3527160*x^18 - 7054320*x^17 + 8817900*x^16 - 7054320*x^15 + 3527160*x^14 - 1007760*x^13 + 125970*x^12
20
13:
-77520*x^20 + 542640*x^19 - 1627920*x^18 + 2713200*x^17 - 2713200*x^16 + 1627920*x^15 - 542640*x^14 + 77520*x^13
20
14:
38760*x^20 - 232560*x^19 + 581400*x^18 - 775200*x^17 + 581400*x^16 - 232560*x^15 + 38760*x^14
20
15:
-15504*x^20 + 77520*x^19 - 155040*x^18 + 155040*x^17 - 77520*x^16 + 15504*x^15
20
16:
4845*x^20 - 19380*x^19 + 29070*x^18 - 19380*x^17 + 4845*x^16
20
17:
-1140*x^20 + 3420*x^19 - 3420*x^18 + 1140*x^17
20
18:
190*x^20 - 380*x^19 + 190*x^18
20
19:
-20*x^20 + 20*x^19
20
20:
x^20
21
0:
-x^21 + 21*x^20 - 210*x^19 + 1330*x^18 - 5985*x^17 + 20349*x^16 - 54264*x^15 + 116280*x^14 - 203490*x^13 + 293930*x^12 - 352716*x^11 + 352716*x^10 - 293930*x^9 + 203490*x^8 - 116280*x^7 + 54264*x^6 - 20349*x^5 + 5985*x^4 - 1330*x^3 + 210*x^2 - 21*x + 1
21
1:
21*x^21 - 420*x^20 + 3990*x^19 - 23940*x^18 + 101745*x^17 - 325584*x^16 + 813960*x^15 - 1627920*x^14 + 2645370*x^13 - 3527160*x^12 + 3879876*x^11 - 3527160*x^10 + 2645370*x^9 - 1627920*x^8 + 813960*x^7 - 325584*x^6 + 101745*x^5 - 23940*x^4 + 3990*x^3 - 420*x^2 + 21*x
21
2:
-210*x^21 + 3990*x^20 - 35910*x^19 + 203490*x^18 - 813960*x^17 + 2441880*x^16 - 5697720*x^15 + 10581480*x^14 - 15872220*x^13 + 19399380*x^12 - 19399380*x^11 + 15872220*x^10 - 10581480*x^9 + 5697720*x^8 - 2441880*x^7 + 813960*x^6 - 203490*x^5 + 35910*x^4 - 3990*x^3 + 210*x^2
21
3:
1330*x^21 - 23940*x^20 + 203490*x^19 - 1085280*x^18 + 4069800*x^17 - 11395440*x^16 + 24690120*x^15 - 42325920*x^14 + 58198140*x^13 - 64664600*x^12 + 58198140*x^11 - 42325920*x^10 + 24690120*x^9 - 11395440*x^8 + 4069800*x^7 - 1085280*x^6 + 203490*x^5 - 23940*x^4 + 1330*x^3
21
4:
-5985*x^21 + 101745*x^20 - 813960*x^19 + 4069800*x^18 - 14244300*x^17 + 37035180*x^16 - 74070360*x^15 + 116396280*x^14 - 145495350*x^13 + 145495350*x^12 - 116396280*x^11 + 74070360*x^10 - 37035180*x^9 + 14244300*x^8 - 4069800*x^7 + 813960*x^6 - 101745*x^5 + 5985*x^4
21
5:
20349*x^21 - 325584*x^20 + 2441880*x^19 - 11395440*x^18 + 37035180*x^17 - 88884432*x^16 + 162954792*x^15 - 232792560*x^14 + 261891630*x^13 - 232792560*x^12 + 162954792*x^11 - 88884432*x^10 + 37035180*x^9 - 11395440*x^8 + 2441880*x^7 - 325584*x^6 + 20349*x^5
21
6:
-54264*x^21 + 813960*x^20 - 5697720*x^19 + 24690120*x^18 - 74070360*x^17 + 162954792*x^16 - 271591320*x^15 + 349188840*x^14 - 349188840*x^13 + 271591320*x^12 - 162954792*x^11 + 74070360*x^10 - 24690120*x^9 + 5697720*x^8 - 813960*x^7 + 54264*x^6
21
7:
116280*x^21 - 1627920*x^20 + 10581480*x^19 - 42325920*x^18 + 116396280*x^17 - 232792560*x^16 + 349188840*x^15 - 399072960*x^14 + 349188840*x^13 - 232792560*x^12 + 116396280*x^11 - 42325920*x^10 + 10581480*x^9 - 1627920*x^8 + 116280*x^7
21
8:
-203490*x^21 + 2645370*x^20 - 15872220*x^19 + 58198140*x^18 - 145495350*x^17 + 261891630*x^16 - 349188840*x^15 + 349188840*x^14 - 261891630*x^13 + 145495350*x^12 - 58198140*x^11 + 15872220*x^10 - 2645370*x^9 + 203490*x^8
21
9:
293930*x^21 - 3527160*x^20 + 19399380*x^19 - 64664600*x^18 + 145495350*x^17 - 232792560*x^16 + 271591320*x^15 - 232792560*x^14 + 145495350*x^13 - 64664600*x^12 + 19399380*x^11 - 3527160*x^10 + 293930*x^9
21
10:
-352716*x^21 + 3879876*x^20 - 19399380*x^19 + 58198140*x^18 - 116396280*x^17 + 162954792*x^16 - 162954792*x^15 + 116396280*x^14 - 58198140*x^13 + 19399380*x^12 - 3879876*x^11 + 352716*x^10
21
11:
352716*x^21 - 3527160*x^20 + 15872220*x^19 - 42325920*x^18 + 74070360*x^17 - 88884432*x^16 + 74070360*x^15 - 42325920*x^14 + 15872220*x^13 - 3527160*x^12 + 352716*x^11
21
12:
-293930*x^21 + 2645370*x^20 - 10581480*x^19 + 24690120*x^18 - 37035180*x^17 + 37035180*x^16 - 24690120*x^15 + 10581480*x^14 - 2645370*x^13 + 293930*x^12
21
13:
203490*x^21 - 1627920*x^20 + 5697720*x^19 - 11395440*x^18 + 14244300*x^17 - 11395440*x^16 + 5697720*x^15 - 1627920*x^14 + 203490*x^13
21
14:
-116280*x^21 + 813960*x^20 - 2441880*x^19 + 4069800*x^18 - 4069800*x^17 + 2441880*x^16 - 813960*x^15 + 116280*x^14
21
15:
54264*x^21 - 325584*x^20 + 813960*x^19 - 1085280*x^18 + 813960*x^17 - 325584*x^16 + 54264*x^15
21
16:
-20349*x^21 + 101745*x^20 - 203490*x^19 + 203490*x^18 - 101745*x^17 + 20349*x^16
21
17:
5985*x^21 - 23940*x^20 + 35910*x^19 - 23940*x^18 + 5985*x^17
21
18:
-1330*x^21 + 3990*x^20 - 3990*x^19 + 1330*x^18
21
19:
210*x^21 - 420*x^20 + 210*x^19
21
20:
-21*x^21 + 21*x^20
21
21:
x^21
22
0:
x^22 - 22*x^21 + 231*x^20 - 1540*x^19 + 7315*x^18 - 26334*x^17 + 74613*x^16 - 170544*x^15 + 319770*x^14 - 497420*x^13 + 646646*x^12 - 705432*x^11 + 646646*x^10 - 497420*x^9 + 319770*x^8 - 170544*x^7 + 74613*x^6 - 26334*x^5 + 7315*x^4 - 1540*x^3 + 231*x^2 - 22*x + 1
22
1:
-22*x^22 + 462*x^21 - 4620*x^20 + 29260*x^19 - 131670*x^18 + 447678*x^17 - 1193808*x^16 + 2558160*x^15 - 4476780*x^14 + 6466460*x^13 - 7759752*x^12 + 7759752*x^11 - 6466460*x^10 + 4476780*x^9 - 2558160*x^8 + 1193808*x^7 - 447678*x^6 + 131670*x^5 - 29260*x^4 + 4620*x^3 - 462*x^2 + 22*x
22
2:
231*x^22 - 4620*x^21 + 43890*x^20 - 263340*x^19 + 1119195*x^18 - 3581424*x^17 + 8953560*x^16 - 17907120*x^15 + 29099070*x^14 - 38798760*x^13 + 42678636*x^12 - 38798760*x^11 + 29099070*x^10 - 17907120*x^9 + 8953560*x^8 - 3581424*x^7 + 1119195*x^6 - 263340*x^5 + 43890*x^4 - 4620*x^3 + 231*x^2
22
3:
-1540*x^22 + 29260*x^21 - 263340*x^20 + 1492260*x^19 - 5969040*x^18 + 17907120*x^17 - 41783280*x^16 + 77597520*x^15 - 116396280*x^14 + 142262120*x^13 - 142262120*x^12 + 116396280*x^11 - 77597520*x^10 + 41783280*x^9 - 17907120*x^8 + 5969040*x^7 - 1492260*x^6 + 263340*x^5 - 29260*x^4 + 1540*x^3
22
4:
7315*x^22 - 131670*x^21 + 1119195*x^20 - 5969040*x^19 + 22383900*x^18 - 62674920*x^17 + 135795660*x^16 - 232792560*x^15 + 320089770*x^14 - 355655300*x^13 + 320089770*x^12 - 232792560*x^11 + 135795660*x^10 - 62674920*x^9 + 22383900*x^8 - 5969040*x^7 + 1119195*x^6 - 131670*x^5 + 7315*x^4
22
5:
-26334*x^22 + 447678*x^21 - 3581424*x^20 + 17907120*x^19 - 62674920*x^18 + 162954792*x^17 - 325909584*x^16 + 512143632*x^15 - 640179540*x^14 + 640179540*x^13 - 512143632*x^12 + 325909584*x^11 - 162954792*x^10 + 62674920*x^9 - 17907120*x^8 + 3581424*x^7 - 447678*x^6 + 26334*x^5
22
6:
74613*x^22 - 1193808*x^21 + 8953560*x^20 - 41783280*x^19 + 135795660*x^18 - 325909584*x^17 + 597500904*x^16 - 853572720*x^15 + 960269310*x^14 - 853572720*x^13 + 597500904*x^12 - 325909584*x^11 + 135795660*x^10 - 41783280*x^9 + 8953560*x^8 - 1193808*x^7 + 74613*x^6
22
7:
-170544*x^22 + 2558160*x^21 - 17907120*x^20 + 77597520*x^19 - 232792560*x^18 + 512143632*x^17 - 853572720*x^16 + 1097450640*x^15 - 1097450640*x^14 + 853572720*x^13 - 512143632*x^12 + 232792560*x^11 - 77597520*x^10 + 17907120*x^9 - 2558160*x^8 + 170544*x^7
22
8:
319770*x^22 - 4476780*x^21 + 29099070*x^20 - 116396280*x^19 + 320089770*x^18 - 640179540*x^17 + 960269310*x^16 - 1097450640*x^15 + 960269310*x^14 - 640179540*x^13 + 320089770*x^12 - 116396280*x^11 + 29099070*x^10 - 4476780*x^9 + 319770*x^8
22
9:
-497420*x^22 + 6466460*x^21 - 38798760*x^20 + 142262120*x^19 - 355655300*x^18 + 640179540*x^17 - 853572720*x^16 + 853572720*x^15 - 640179540*x^14 + 355655300*x^13 - 142262120*x^12 + 38798760*x^11 - 6466460*x^10 + 497420*x^9
22
10:
646646*x^22 - 7759752*x^21 + 42678636*x^20 - 142262120*x^19 + 320089770*x^18 - 512143632*x^17 + 597500904*x^16 - 512143632*x^15 + 320089770*x^14 - 142262120*x^13 + 42678636*x^12 - 7759752*x^11 + 646646*x^10
22
11:
-705432*x^22 + 7759752*x^21 - 38798760*x^20 + 116396280*x^19 - 232792560*x^18 + 325909584*x^17 - 325909584*x^16 + 232792560*x^15 - 116396280*x^14 + 38798760*x^13 - 7759752*x^12 + 705432*x^11
22
12:
646646*x^22 - 6466460*x^21 + 29099070*x^20 - 77597520*x^19 + 135795660*x^18 - 162954792*x^17 + 135795660*x^16 - 77597520*x^15 + 29099070*x^14 - 6466460*x^13 + 646646*x^12
22
13:
-497420*x^22 + 4476780*x^21 - 17907120*x^20 + 41783280*x^19 - 62674920*x^18 + 62674920*x^17 - 41783280*x^16 + 17907120*x^15 - 4476780*x^14 + 497420*x^13
22
14:
319770*x^22 - 2558160*x^21 + 8953560*x^20 - 17907120*x^19 + 22383900*x^18 - 17907120*x^17 + 8953560*x^16 - 2558160*x^15 + 319770*x^14
22
15:
-170544*x^22 + 1193808*x^21 - 3581424*x^20 + 5969040*x^19 - 5969040*x^18 + 3581424*x^17 - 1193808*x^16 + 170544*x^15
22
16:
74613*x^22 - 447678*x^21 + 1119195*x^20 - 1492260*x^19 + 1119195*x^18 - 447678*x^17 + 74613*x^16
22
17:
-26334*x^22 + 131670*x^21 - 263340*x^20 + 263340*x^19 - 131670*x^18 + 26334*x^17
22
18:
7315*x^22 - 29260*x^21 + 43890*x^20 - 29260*x^19 + 7315*x^18
22
19:
-1540*x^22 + 4620*x^21 - 4620*x^20 + 1540*x^19
22
20:
231*x^22 - 462*x^21 + 231*x^20
22
21:
-22*x^22 + 22*x^21
22
22:
x^22
23
0:
-x^23 + 23*x^22 - 253*x^21 + 1771*x^20 - 8855*x^19 + 33649*x^18 - 100947*x^17 + 245157*x^16 - 490314*x^15 + 817190*x^14 - 1144066*x^13 + 1352078*x^12 - 1352078*x^11 + 1144066*x^10 - 817190*x^9 + 490314*x^8 - 245157*x^7 + 100947*x^6 - 33649*x^5 + 8855*x^4 - 1771*x^3 + 253*x^2 - 23*x + 1
23
1:
23*x^23 - 506*x^22 + 5313*x^21 - 35420*x^20 + 168245*x^19 - 605682*x^18 + 1716099*x^17 - 3922512*x^16 + 7354710*x^15 - 11440660*x^14 + 14872858*x^13 - 16224936*x^12 + 14872858*x^11 - 11440660*x^10 + 7354710*x^9 - 3922512*x^8 + 1716099*x^7 - 605682*x^6 + 168245*x^5 - 35420*x^4 + 5313*x^3 - 506*x^2 + 23*x
23
2:
-253*x^23 + 5313*x^22 - 53130*x^21 + 336490*x^20 - 1514205*x^19 + 5148297*x^18 - 13728792*x^17 + 29418840*x^16 - 51482970*x^15 + 74364290*x^14 - 89237148*x^13 + 89237148*x^12 - 74364290*x^11 + 51482970*x^10 - 29418840*x^9 + 13728792*x^8 - 5148297*x^7 + 1514205*x^6 - 336490*x^5 + 53130*x^4 - 5313*x^3 + 253*x^2
23
3:
1771*x^23 - 35420*x^22 + 336490*x^21 - 2018940*x^20 + 8580495*x^19 - 27457584*x^18 + 68643960*x^17 - 137287920*x^16 + 223092870*x^15 - 297457160*x^14 + 327202876*x^13 - 297457160*x^12 + 223092870*x^11 - 137287920*x^10 + 68643960*x^9 - 27457584*x^8 + 8580495*x^7 - 2018940*x^6 + 336490*x^5 - 35420*x^4 + 1771*x^3
23
4:
-8855*x^23 + 168245*x^22 - 1514205*x^21 + 8580495*x^20 - 34321980*x^19 + 102965940*x^18 - 240253860*x^17 + 446185740*x^16 - 669278610*x^15 + 818007190*x^14 - 818007190*x^13 + 669278610*x^12 - 446185740*x^11 + 240253860*x^10 - 102965940*x^9 + 34321980*x^8 - 8580495*x^7 + 1514205*x^6 - 168245*x^5 + 8855*x^4
23
5:
33649*x^23 - 605682*x^22 + 5148297*x^21 - 27457584*x^20 + 102965940*x^19 - 288304632*x^18 + 624660036*x^17 - 1070845776*x^16 + 1472412942*x^15 - 1636014380*x^14 + 1472412942*x^13 - 1070845776*x^12 + 624660036*x^11 - 288304632*x^10 + 102965940*x^9 - 27457584*x^8 + 5148297*x^7 - 605682*x^6 + 33649*x^5
23
6:
-100947*x^23 + 1716099*x^22 - 13728792*x^21 + 68643960*x^20 - 240253860*x^19 + 624660036*x^18 - 1249320072*x^17 + 1963217256*x^16 - 2454021570*x^15 + 2454021570*x^14 - 1963217256*x^13 + 1249320072*x^12 - 624660036*x^11 + 240253860*x^10 - 68643960*x^9 + 13728792*x^8 - 1716099*x^7 + 100947*x^6
23
7:
245157*x^23 - 3922512*x^22 + 29418840*x^21 - 137287920*x^20 + 446185740*x^19 - 1070845776*x^18 + 1963217256*x^17 - 2804596080*x^16 + 3155170590*x^15 - 2804596080*x^14 + 1963217256*x^13 - 1070845776*x^12 + 446185740*x^11 - 137287920*x^10 + 29418840*x^9 - 3922512*x^8 + 245157*x^7
23
8:
-490314*x^23 + 7354710*x^22 - 51482970*x^21 + 223092870*x^20 - 669278610*x^19 + 1472412942*x^18 - 2454021570*x^17 + 3155170590*x^16 - 3155170590*x^15 + 2454021570*x^14 - 1472412942*x^13 + 669278610*x^12 - 223092870*x^11 + 51482970*x^10 - 7354710*x^9 + 490314*x^8
23
9:
817190*x^23 - 11440660*x^22 + 74364290*x^21 - 297457160*x^20 + 818007190*x^19 - 1636014380*x^18 + 2454021570*x^17 - 2804596080*x^16 + 2454021570*x^15 - 1636014380*x^14 + 818007190*x^13 - 297457160*x^12 + 74364290*x^11 - 11440660*x^10 + 817190*x^9
23
10:
-1144066*x^23 + 14872858*x^22 - 89237148*x^21 + 327202876*x^20 - 818007190*x^19 + 1472412942*x^18 - 1963217256*x^17 + 1963217256*x^16 - 1472412942*x^15 + 818007190*x^14 - 327202876*x^13 + 89237148*x^12 - 14872858*x^11 + 1144066*x^10
23
11:
1352078*x^23 - 16224936*x^22 + 89237148*x^21 - 297457160*x^20 + 669278610*x^19 - 1070845776*x^18 + 1249320072*x^17 - 1070845776*x^16 + 669278610*x^15 - 297457160*x^14 + 89237148*x^13 - 16224936*x^12 + 1352078*x^11
23
12:
-1352078*x^23 + 14872858*x^22 - 74364290*x^21 + 223092870*x^20 - 446185740*x^19 + 624660036*x^18 - 624660036*x^17 + 446185740*x^16 - 223092870*x^15 + 74364290*x^14 - 14872858*x^13 + 1352078*x^12
23
13:
1144066*x^23 - 11440660*x^22 + 51482970*x^21 - 137287920*x^20 + 240253860*x^19 - 288304632*x^18 + 240253860*x^17 - 137287920*x^16 + 51482970*x^15 - 11440660*x^14 + 1144066*x^13
23
14:
-817190*x^23 + 7354710*x^22 - 29418840*x^21 + 68643960*x^20 - 102965940*x^19 + 102965940*x^18 - 68643960*x^17 + 29418840*x^16 - 7354710*x^15 + 817190*x^14
23
15:
490314*x^23 - 3922512*x^22 + 13728792*x^21 - 27457584*x^20 + 34321980*x^19 - 27457584*x^18 + 13728792*x^17 - 3922512*x^16 + 490314*x^15
23
16:
-245157*x^23 + 1716099*x^22 - 5148297*x^21 + 8580495*x^20 - 8580495*x^19 + 5148297*x^18 - 1716099*x^17 + 245157*x^16
23
17:
100947*x^23 - 605682*x^22 + 1514205*x^21 - 2018940*x^20 + 1514205*x^19 - 605682*x^18 + 100947*x^17
23
18:
-33649*x^23 + 168245*x^22 - 336490*x^21 + 336490*x^20 - 168245*x^19 + 33649*x^18
23
19:
8855*x^23 - 35420*x^22 + 53130*x^21 - 35420*x^20 + 8855*x^19
23
20:
-1771*x^23 + 5313*x^22 - 5313*x^21 + 1771*x^20
23
21:
253*x^23 - 506*x^22 + 253*x^21
23
22:
-23*x^23 + 23*x^22
23
23:
x^23
24
0:
x^24 - 24*x^23 + 276*x^22 - 2024*x^21 + 10626*x^20 - 42504*x^19 + 134596*x^18 - 346104*x^17 + 735471*x^16 - 1307504*x^15 + 1961256*x^14 - 2496144*x^13 + 2704156*x^12 - 2496144*x^11 + 1961256*x^10 - 1307504*x^9 + 735471*x^8 - 346104*x^7 + 134596*x^6 - 42504*x^5 + 10626*x^4 - 2024*x^3 + 276*x^2 - 24*x + 1
24
1:
-24*x^24 + 552*x^23 - 6072*x^22 + 42504*x^21 - 212520*x^20 + 807576*x^19 - 2422728*x^18 + 5883768*x^17 - 11767536*x^16 + 19612560*x^15 - 27457584*x^14 + 32449872*x^13 - 32449872*x^12 + 27457584*x^11 - 19612560*x^10 + 11767536*x^9 - 5883768*x^8 + 2422728*x^7 - 807576*x^6 + 212520*x^5 - 42504*x^4 + 6072*x^3 - 552*x^2 + 24*x
24
2:
276*x^24 - 6072*x^23 + 63756*x^22 - 425040*x^21 + 2018940*x^20 - 7268184*x^19 + 20593188*x^18 - 47070144*x^17 + 88256520*x^16 - 137287920*x^15 + 178474296*x^14 - 194699232*x^13 + 178474296*x^12 - 137287920*x^11 + 88256520*x^10 - 47070144*x^9 + 20593188*x^8 - 7268184*x^7 + 2018940*x^6 - 425040*x^5 + 63756*x^4 - 6072*x^3 + 276*x^2
24
3:
-2024*x^24 + 42504*x^23 - 425040*x^22 + 2691920*x^21 - 12113640*x^20 + 41186376*x^19 - 109830336*x^18 + 235350720*x^17 - 411863760*x^16 + 594914320*x^15 - 713897184*x^14 + 713897184*x^13 - 594914320*x^12 + 411863760*x^11 - 235350720*x^10 + 109830336*x^9 - 41186376*x^8 + 12113640*x^7 - 2691920*x^6 + 425040*x^5 - 42504*x^4 + 2024*x^3
24
4:
10626*x^24 - 212520*x^23 + 2018940*x^22 - 12113640*x^21 + 51482970*x^20 - 164745504*x^19 + 411863760*x^18 - 823727520*x^17 + 1338557220*x^16 - 1784742960*x^15 + 1963217256*x^14 - 1784742960*x^13 + 1338557220*x^12 - 823727520*x^11 + 411863760*x^10 - 164745504*x^9 + 51482970*x^8 - 12113640*x^7 + 2018940*x^6 - 212520*x^5 + 10626*x^4
24
5:
-42504*x^24 + 807576*x^23 - 7268184*x^22 + 41186376*x^21 - 164745504*x^20 + 494236512*x^19 - 1153218528*x^18 + 2141691552*x^17 - 3212537328*x^16 + 3926434512*x^15 - 3926434512*x^14 + 3212537328*x^13 - 2141691552*x^12 + 1153218528*x^11 - 494236512*x^10 + 164745504*x^9 - 41186376*x^8 + 7268184*x^7 - 807576*x^6 + 42504*x^5
24
6:
134596*x^24 - 2422728*x^23 + 20593188*x^22 - 109830336*x^21 + 411863760*x^20 - 1153218528*x^19 + 2498640144*x^18 - 4283383104*x^17 + 5889651768*x^16 - 6544057520*x^15 + 5889651768*x^14 - 4283383104*x^13 + 2498640144*x^12 - 1153218528*x^11 + 411863760*x^10 - 109830336*x^9 + 20593188*x^8 - 2422728*x^7 + 134596*x^6
24
7:
-346104*x^24 + 5883768*x^23 - 47070144*x^22 + 235350720*x^21 - 823727520*x^20 + 2141691552*x^19 - 4283383104*x^18 + 6731030592*x^17 - 8413788240*x^16 + 8413788240*x^15 - 6731030592*x^14 + 4283383104*x^13 - 2141691552*x^12 + 823727520*x^11 - 235350720*x^10 + 47070144*x^9 - 5883768*x^8 + 346104*x^7
24
8:
735471*x^24 - 11767536*x^23 + 88256520*x^22 - 411863760*x^21 + 1338557220*x^20 - 3212537328*x^19 + 5889651768*x^18 - 8413788240*x^17 + 9465511770*x^16 - 8413788240*x^15 + 5889651768*x^14 - 3212537328*x^13 + 1338557220*x^12 - 411863760*x^11 + 88256520*x^10 - 11767536*x^9 + 735471*x^8
24
9:
-1307504*x^24 + 19612560*x^23 - 137287920*x^22 + 594914320*x^21 - 1784742960*x^20 + 3926434512*x^19 - 6544057520*x^18 + 8413788240*x^17 - 8413788240*x^16 + 6544057520*x^15 - 3926434512*x^14 + 1784742960*x^13 - 594914320*x^12 + 137287920*x^11 - 19612560*x^10 + 1307504*x^9
24
10:
1961256*x^24 - 27457584*x^23 + 178474296*x^22 - 713897184*x^21 + 1963217256*x^20 - 3926434512*x^19 + 5889651768*x^18 - 6731030592*x^17 + 5889651768*x^16 - 3926434512*x^15 + 1963217256*x^14 - 713897184*x^13 + 178474296*x^12 - 27457584*x^11 + 1961256*x^10
24
11:
-2496144*x^24 + 32449872*x^23 - 194699232*x^22 + 713897184*x^21 - 1784742960*x^20 + 3212537328*x^19 - 4283383104*x^18 + 4283383104*x^17 - 3212537328*x^16 + 1784742960*x^15 - 713897184*x^14 + 194699232*x^13 - 32449872*x^12 + 2496144*x^11
24
12:
2704156*x^24 - 32449872*x^23 + 178474296*x^22 - 594914320*x^21 + 1338557220*x^20 - 2141691552*x^19 + 2498640144*x^18 - 2141691552*x^17 + 1338557220*x^16 - 594914320*x^15 + 178474296*x^14 - 32449872*x^13 + 2704156*x^12
24
13:
-2496144*x^24 + 27457584*x^23 - 137287920*x^22 + 411863760*x^21 - 823727520*x^20 + 1153218528*x^19 - 1153218528*x^18 + 823727520*x^17 - 411863760*x^16 + 137287920*x^15 - 27457584*x^14 + 2496144*x^13
24
14:
1961256*x^24 - 19612560*x^23 + 88256520*x^22 - 235350720*x^21 + 411863760*x^20 - 494236512*x^19 + 411863760*x^18 - 235350720*x^17 + 88256520*x^16 - 19612560*x^15 + 1961256*x^14
24
15:
-1307504*x^24 + 11767536*x^23 - 47070144*x^22 + 109830336*x^21 - 164745504*x^20 + 164745504*x^19 - 109830336*x^18 + 47070144*x^17 - 11767536*x^16 + 1307504*x^15
24
16:
735471*x^24 - 5883768*x^23 + 20593188*x^22 - 41186376*x^21 + 51482970*x^20 - 41186376*x^19 + 20593188*x^18 - 5883768*x^17 + 735471*x^16
24
17:
-346104*x^24 + 2422728*x^23 - 7268184*x^22 + 12113640*x^21 - 12113640*x^20 + 7268184*x^19 - 2422728*x^18 + 346104*x^17
24
18:
134596*x^24 - 807576*x^23 + 2018940*x^22 - 2691920*x^21 + 2018940*x^20 - 807576*x^19 + 134596*x^18
24
19:
-42504*x^24 + 212520*x^23 - 425040*x^22 + 425040*x^21 - 212520*x^20 + 42504*x^19
24
20:
10626*x^24 - 42504*x^23 + 63756*x^22 - 42504*x^21 + 10626*x^20
24
21:
-2024*x^24 + 6072*x^23 - 6072*x^22 + 2024*x^21
24
22:
276*x^24 - 552*x^23 + 276*x^22
24
23:
-24*x^24 + 24*x^23
24
24:
x^24
25
0:
-x^25 + 25*x^24 - 300*x^23 + 2300*x^22 - 12650*x^21 + 53130*x^20 - 177100*x^19 + 480700*x^18 - 1081575*x^17 + 2042975*x^16 - 3268760*x^15 + 4457400*x^14 - 5200300*x^13 + 5200300*x^12 - 4457400*x^11 + 3268760*x^10 - 2042975*x^9 + 1081575*x^8 - 480700*x^7 + 177100*x^6 - 53130*x^5 + 12650*x^4 - 2300*x^3 + 300*x^2 - 25*x + 1
25
1:
25*x^25 - 600*x^24 + 6900*x^23 - 50600*x^22 + 265650*x^21 - 1062600*x^20 + 3364900*x^19 - 8652600*x^18 + 18386775*x^17 - 32687600*x^16 + 49031400*x^15 - 62403600*x^14 + 67603900*x^13 - 62403600*x^12 + 49031400*x^11 - 32687600*x^10 + 18386775*x^9 - 8652600*x^8 + 3364900*x^7 - 1062600*x^6 + 265650*x^5 - 50600*x^4 + 6900*x^3 - 600*x^2 + 25*x
25
2:
-300*x^25 + 6900*x^24 - 75900*x^23 + 531300*x^22 - 2656500*x^21 + 10094700*x^20 - 30284100*x^19 + 73547100*x^18 - 147094200*x^17 + 245157000*x^16 - 343219800*x^15 + 405623400*x^14 - 405623400*x^13 + 343219800*x^12 - 245157000*x^11 + 147094200*x^10 - 73547100*x^9 + 30284100*x^8 - 10094700*x^7 + 2656500*x^6 - 531300*x^5 + 75900*x^4 - 6900*x^3 + 300*x^2
25
3:
2300*x^25 - 50600*x^24 + 531300*x^23 - 3542000*x^22 + 16824500*x^21 - 60568200*x^20 + 171609900*x^19 - 392251200*x^18 + 735471000*x^17 - 1144066000*x^16 + 1487285800*x^15 - 1622493600*x^14 + 1487285800*x^13 - 1144066000*x^12 + 735471000*x^11 - 392251200*x^10 + 171609900*x^9 - 60568200*x^8 + 16824500*x^7 - 3542000*x^6 + 531300*x^5 - 50600*x^4 + 2300*x^3
25
4:
-12650*x^25 + 265650*x^24 - 2656500*x^23 + 16824500*x^22 - 75710250*x^21 + 257414850*x^20 - 686439600*x^19 + 1470942000*x^18 - 2574148500*x^17 + 3718214500*x^16 - 4461857400*x^15 + 4461857400*x^14 - 3718214500*x^13 + 2574148500*x^12 - 1470942000*x^11 + 686439600*x^10 - 257414850*x^9 + 75710250*x^8 - 16824500*x^7 + 2656500*x^6 - 265650*x^5 + 12650*x^4
25
5:
53130*x^25 - 1062600*x^24 + 10094700*x^23 - 60568200*x^22 + 257414850*x^21 - 823727520*x^20 + 2059318800*x^19 - 4118637600*x^18 + 6692786100*x^17 - 8923714800*x^16 + 9816086280*x^15 - 8923714800*x^14 + 6692786100*x^13 - 4118637600*x^12 + 2059318800*x^11 - 823727520*x^10 + 257414850*x^9 - 60568200*x^8 + 10094700*x^7 - 1062600*x^6 + 53130*x^5
25
6:
-177100*x^25 + 3364900*x^24 - 30284100*x^23 + 171609900*x^22 - 686439600*x^21 + 2059318800*x^20 - 4805077200*x^19 + 8923714800*x^18 - 13385572200*x^17 + 16360143800*x^16 - 16360143800*x^15 + 13385572200*x^14 - 8923714800*x^13 + 4805077200*x^12 - 2059318800*x^11 + 686439600*x^10 - 171609900*x^9 + 30284100*x^8 - 3364900*x^7 + 177100*x^6
25
7:
480700*x^25 - 8652600*x^24 + 73547100*x^23 - 392251200*x^22 + 1470942000*x^21 - 4118637600*x^20 + 8923714800*x^19 - 15297796800*x^18 + 21034470600*x^17 - 23371634000*x^16 + 21034470600*x^15 - 15297796800*x^14 + 8923714800*x^13 - 4118637600*x^12 + 1470942000*x^11 - 392251200*x^10 + 73547100*x^9 - 8652600*x^8 + 480700*x^7
25
8:
-1081575*x^25 + 18386775*x^24 - 147094200*x^23 + 735471000*x^22 - 2574148500*x^21 + 6692786100*x^20 - 13385572200*x^19 + 21034470600*x^18 - 26293088250*x^17 + 26293088250*x^16 - 21034470600*x^15 + 13385572200*x^14 - 6692786100*x^13 + 2574148500*x^12 - 735471000*x^11 + 147094200*x^10 - 18386775*x^9 + 1081575*x^8
25
9:
2042975*x^25 - 32687600*x^24 + 245157000*x^23 - 1144066000*x^22 + 3718214500*x^21 - 8923714800*x^20 + 16360143800*x^19 - 23371634000*x^18 + 26293088250*x^17 - 23371634000*x^16 + 16360143800*x^15 - 8923714800*x^14 + 3718214500*x^13 - 1144066000*x^12 + 245157000*x^11 - 32687600*x^10 + 2042975*x^9
25
10:
-3268760*x^25 + 49031400*x^24 - 343219800*x^23 + 1487285800*x^22 - 4461857400*x^21 + 9816086280*x^20 - 16360143800*x^19 + 21034470600*x^18 - 21034470600*x^17 + 16360143800*x^16 - 9816086280*x^15 + 4461857400*x^14 - 1487285800*x^13 + 343219800*x^12 - 49031400*x^11 + 3268760*x^10
25
11:
4457400*x^25 - 62403600*x^24 + 405623400*x^23 - 1622493600*x^22 + 4461857400*x^21 - 8923714800*x^20 + 13385572200*x^19 - 15297796800*x^18 + 13385572200*x^17 - 8923714800*x^16 + 4461857400*x^15 - 1622493600*x^14 + 405623400*x^13 - 62403600*x^12 + 4457400*x^11
25
12:
-5200300*x^25 + 67603900*x^24 - 405623400*x^23 + 1487285800*x^22 - 3718214500*x^21 + 6692786100*x^20 - 8923714800*x^19 + 8923714800*x^18 - 6692786100*x^17 + 3718214500*x^16 - 1487285800*x^15 + 405623400*x^14 - 67603900*x^13 + 5200300*x^12
25
13:
5200300*x^25 - 62403600*x^24 + 343219800*x^23 - 1144066000*x^22 + 2574148500*x^21 - 4118637600*x^20 + 4805077200*x^19 - 4118637600*x^18 + 2574148500*x^17 - 1144066000*x^16 + 343219800*x^15 - 62403600*x^14 + 5200300*x^13
25
14:
-4457400*x^25 + 49031400*x^24 - 245157000*x^23 + 735471000*x^22 - 1470942000*x^21 + 2059318800*x^20 - 2059318800*x^19 + 1470942000*x^18 - 735471000*x^17 + 245157000*x^16 - 49031400*x^15 + 4457400*x^14
25
15:
3268760*x^25 - 32687600*x^24 + 147094200*x^23 - 392251200*x^22 + 686439600*x^21 - 823727520*x^20 + 686439600*x^19 - 392251200*x^18 + 147094200*x^17 - 32687600*x^16 + 3268760*x^15
25
16:
-2042975*x^25 + 18386775*x^24 - 73547100*x^23 + 171609900*x^22 - 257414850*x^21 + 257414850*x^20 - 171609900*x^19 + 73547100*x^18 - 18386775*x^17 + 2042975*x^16
25
17:
1081575*x^25 - 8652600*x^24 + 30284100*x^23 - 60568200*x^22 + 75710250*x^21 - 60568200*x^20 + 30284100*x^19 - 8652600*x^18 + 1081575*x^17
25
18:
-480700*x^25 + 3364900*x^24 - 10094700*x^23 + 16824500*x^22 - 16824500*x^21 + 10094700*x^20 - 3364900*x^19 + 480700*x^18
25
19:
177100*x^25 - 1062600*x^24 + 2656500*x^23 - 3542000*x^22 + 2656500*x^21 - 1062600*x^20 + 177100*x^19
25
20:
-53130*x^25 + 265650*x^24 - 531300*x^23 + 531300*x^22 - 265650*x^21 + 53130*x^20
25
21:
12650*x^25 - 50600*x^24 + 75900*x^23 - 50600*x^22 + 12650*x^21
25
22:
-2300*x^25 + 6900*x^24 - 6900*x^23 + 2300*x^22
25
23:
300*x^25 - 600*x^24 + 300*x^23
25
24:
-25*x^25 + 25*x^24
25
25:
x^25
Definition
For integers $0 \leq v \leq n$ the Bernstein basis polynomial is $b_{v,n}(x) = \binom{n}{v} x^{v} (1-x)^{n-v}$. Parameters are given as $n$ then $v$, the degree before the index, though the usual notation writes the index first.
Parameters
$n$
—   integer ($0 \leq n \leq 25$)
$v$
—   integer ($0 \leq v \leq n$)
Formulas
(1)
$b_{v,n}(x) = \binom{n}{v} x^{v} (1-x)^{n-v}$.
(2)
$\sum_{v=0}^{n} b_{v,n}(x) = 1$ for every $n$, which is why a Bezier curve lies in the convex hull of its control points.
(3)
$b_{v,n}(x) = (1-x)\,b_{v,n-1}(x) + x\,b_{v-1,n-1}(x)$, reading $b_{v,n-1} = 0$ when $v > n-1$ and $b_{-1,n-1} = 0$.
(4)
$b_{v,n}$ has degree $n$, a zero of order $v$ at $x = 0$ and a zero of order $n-v$ at $x = 1$, and no others.
(5)
$b_{v,n}'(x) = n\left(b_{v-1,n-1}(x) - b_{v,n-1}(x)\right)$.
Comments
(6)
Stored expanded, as a polynomial in $x$ with integer coefficients, rather than in the factored form of the definition: that is what makes two polynomials comparable here.
(7)
The Bernstein polynomial of a function $f$, $\sum_{v} f(v/n)\, b_{v,n}$, converges uniformly to $f$ on $[0,1]$ for continuous $f$ -- Bernstein's proof of the Weierstrass approximation theorem [1].
(8)
The $n+1$ polynomials $b_{0,n}, \dots, b_{n,n}$ are a basis of the polynomials of degree at most $n$, and are the basis in which a B\'ezier curve of degree $n$ is written.
(9)
A different basis of the same space: the Lagrange basis Lagrange_basis_polynomials_for_equally_spaced_nodes also spans the polynomials of degree at most $n$, and is the one to use when the values at the nodes are what is known. These are the basis to use when the coefficients are to be controlled, since they are non-negative on $[0,1]$ and sum to one.
Programs
(P1)
Sage
R. = ZZ[]
def bernstein_basis(v, n):
    return binomial(n, v) * x^v * (1 - x)^(n - v)

bernstein_basis(0, 26)           # the next degree after this table
Links
Data properties
Entries are of type: integral polynomial
Table is complete: false
How well the digits are known: exact


API Python package Skill for AI Privacy Legal notice Source
Data licensed CC BY-SA 4.0. No tracking, no analytics, essential cookies only.