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Adams–Bashforth coefficients
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Numbers
$s$
$j$ 
$\beta_{s,j}$
1
0:
1
comment: This is the forward Euler method.
2
0:
-1/2
comment: 2-step Adams-Bashforth formula, from newest to oldest: $\tfrac{1}{2}(3,-1)$.
2
1:
3/2
3
0:
5/12
comment: 3-step Adams-Bashforth formula, from newest to oldest: $\tfrac{1}{12}(23,-16,5)$.
3
1:
-4/3
3
2:
23/12
4
0:
-3/8
comment: 4-step Adams-Bashforth formula, from newest to oldest: $\tfrac{1}{24}(55,-59,37,-9)$.
4
1:
37/24
4
2:
-59/24
4
3:
55/24
5
0:
251/720
comment: 5-step Adams-Bashforth formula, from newest to oldest: $\tfrac{1}{720}(1901,-2774,2616,-1274,251)$.
5
1:
-637/360
5
2:
109/30
5
3:
-1387/360
5
4:
1901/720
6
0:
-95/288
comment: 6-step Adams-Bashforth formula, from newest to oldest: $\tfrac{1}{1440}(4277,-7923,9982,-7298,2877,-475)$.
6
1:
959/480
6
2:
-3649/720
6
3:
4991/720
6
4:
-2641/480
6
5:
4277/1440
7
0:
19087/60480
comment: 7-step Adams-Bashforth formula, from newest to oldest: $\tfrac{1}{60480}(198721,-447288,705549,-688256,407139,-134472,19087)$.
7
1:
-5603/2520
7
2:
135713/20160
7
3:
-10754/945
7
4:
235183/20160
7
5:
-18637/2520
7
6:
198721/60480
8
0:
-5257/17280
comment: 8-step Adams-Bashforth formula, from newest to oldest: $\tfrac{1}{120960}(434241,-1152169,2183877,-2664477,2102243,-1041723,295767,-36799)$.
8
1:
32863/13440
8
2:
-115747/13440
8
3:
2102243/120960
8
4:
-296053/13440
8
5:
242653/13440
8
6:
-1152169/120960
8
7:
16083/4480
9
0:
1070017/3628800
comment: 9-step Adams-Bashforth formula, from newest to oldest: $\tfrac{1}{3628800}(14097247,-43125206,95476786,-139855262,137968480,-91172642,38833486,-9664106,1070017)$.
9
1:
-4832053/1814400
9
2:
19416743/1814400
9
3:
-45586321/1814400
9
4:
862303/22680
9
5:
-69927631/1814400
9
6:
47738393/1814400
9
7:
-21562603/1814400
9
8:
14097247/3628800
10
0:
-25713/89600
comment: 10-step Adams-Bashforth formula, from newest to oldest: $\tfrac{1}{7257600}(30277247,-104995189,265932680,-454661776,538363838,-444772162,252618224,-94307320,20884811,-2082753)$.
10
1:
20884811/7257600
10
2:
-2357683/181440
10
3:
15788639/453600
10
4:
-222386081/3628800
10
5:
269181919/3628800
10
6:
-28416361/453600
10
7:
6648317/181440
10
8:
-104995189/7257600
10
9:
4325321/1036800
11
0:
26842253/95800320
comment: 11-step Adams-Bashforth formula, from newest to oldest: $\tfrac{1}{479001600}(2132509567,-8271795124,23591063805,-46113029016,63716378958,-63176201472,44857168434,-22329634920,7417904451,-1479574348,134211265)$.
11
1:
-52841941/17107200
11
2:
2472634817/159667200
11
3:
-186080291/3991680
11
4:
2492064913/26611200
11
5:
-82260679/623700
11
6:
3539798831/26611200
11
7:
-1921376209/19958400
11
8:
1572737587/31933440
11
9:
-2067948781/119750400
11
10:
2132509567/479001600
12
0:
-4777223/17418240
comment: 12-step Adams-Bashforth formula, from newest to oldest: $\tfrac{1}{958003200}(4527766399,-19433810163,61633227185,-135579356757,214139355366,-247741639374,211103573298,-131365867290,58189107627,-17410248271,3158642445,-262747265)$.
12
1:
30082309/9123840
12
2:
-17410248271/958003200
12
3:
923636629/15206400
12
4:
-625551749/4561920
12
5:
35183928883/159667200
12
6:
-41290273229/159667200
12
7:
35689892561/159667200
12
8:
-15064372973/106444800
12
9:
12326645437/191600640
12
10:
-6477936721/319334400
12
11:
4527766399/958003200
Definition
For $s\geq1$, the Adams–Bashforth coefficients $\beta_{s,j}$ are the rational numbers for which the explicit $s$-step formula $y_{n+s}=y_{n+s-1}+h\sum_{j=0}^{s-1}\beta_{s,j}f_{n+j}$ is exact whenever $f(t,y(t))$ is a polynomial in $t$ of degree at most $s-1$ [1].
Parameters
$s$
—   number of steps ($s\geq1$)
$j$
—   history index, counted from the oldest value ($0\leq j<s$)
Formulas
(1)
$\beta_{s,j}=\int_{s-1}^{s}\ell_{s-1,j}(x)\,\mathrm{d}x$, where $\ell_{s-1,j}$ is the Lagrange basis polynomial for the nodes $0,1,\ldots,s-1$, so $\ell_{s-1,j}(i)=1$ for $i=j$ and $0$ for the other nodes.
(2)
$\sum_{j=0}^{s-1}\beta_{s,j}j^m=(s^{m+1}-(s-1)^{m+1})/(m+1)$ for $0\leq m\leq s-1$.
Comments
(3)
Here $f_{n+j}=f(t_{n+j},y_{n+j})$ and $t_{n+j}=t_0+(n+j)h$.
(4)
The step size is factored out, so the coefficient of $f_{n+j}$ in the formula is $h\beta_{s,j}$.
(5)
Entries are listed from the oldest value to the newest value. Wikipedia [1] writes the displayed formulas from newest to oldest; for example its three-step row is $(23,-16,5)/12$, while this table stores $j=0,1,2$ as $5/12,-4/3,23/12$.
Programs
(P1)
Sage
import numberdb.sage as numberdb
from sage.matrix.constructor import matrix
from sage.modules.free_module_element import vector
from sage.rings.rational_field import QQ

def adams_bashforth_coefficients(s):
    nodes = list(range(s))
    rows = [[QQ(j) ** m for j in nodes] for m in range(s)]
    rhs = [(QQ(s) ** (m + 1) - QQ(s - 1) ** (m + 1)) / QQ(m + 1)
           for m in range(s)]
    return dict(zip(nodes, matrix(QQ, rows).solve_right(vector(QQ, rhs))))

print(adams_bashforth_coefficients(3))
Links
Similar tables
Lagrange basis polynomials for equally spaced nodes —   each $\beta_{s,j}$ is the integral of $\ell_{s-1,j}$ over $[s-1,s]$
Newton–Cotes weights —   integrals of the same basis over the interval spanned by the interpolation nodes rather than over the next step
Data properties
Entries are of type: rational number
How well the digits are known: exact
Table is complete: no (it holds every coefficient with $1\leq s\leq12$, matching VODE's default maximum Adams order [2])
How they were obtained:

The generator builds the moment equations over $\mathbb{Q}$ for the nodes $0,1,\ldots,s-1$ and solves them exactly. Each completed method is checked against the defining moments, the integral formula for the Lagrange basis polynomials, and the displayed Adams–Bashforth formulas through $s=5$ in [1].



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