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Adams–Moulton coefficients
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Numbers
$s$
$j$ 
$\beta^*_{s,j}$
0
0:
1
comment: This is the backward Euler method.
1
0:
1/2
comment: Trapezoidal rule, from newest to oldest: $\tfrac{1}{2}(1,1)$.
1
1:
1/2
2
0:
-1/12
comment: 2-step Adams-Moulton formula, from newest to oldest: $\tfrac{1}{12}(5,8,-1)$.
2
1:
2/3
2
2:
5/12
3
0:
1/24
comment: 3-step Adams-Moulton formula, from newest to oldest: $\tfrac{1}{24}(9,19,-5,1)$.
3
1:
-5/24
3
2:
19/24
3
3:
3/8
4
0:
-19/720
comment: 4-step Adams-Moulton formula, from newest to oldest: $\tfrac{1}{720}(251,646,-264,106,-19)$.
4
1:
53/360
4
2:
-11/30
4
3:
323/360
4
4:
251/720
5
0:
3/160
comment: 5-step Adams-Moulton formula, from newest to oldest: $\tfrac{1}{1440}(475,1427,-798,482,-173,27)$.
5
1:
-173/1440
5
2:
241/720
5
3:
-133/240
5
4:
1427/1440
5
5:
95/288
6
0:
-863/60480
comment: 6-step Adams-Moulton formula, from newest to oldest: $\tfrac{1}{60480}(19087,65112,-46461,37504,-20211,6312,-863)$.
6
1:
263/2520
6
2:
-6737/20160
6
3:
586/945
6
4:
-15487/20160
6
5:
2713/2520
6
6:
19087/60480
7
0:
275/24192
comment: 7-step Adams-Moulton formula, from newest to oldest: $\tfrac{1}{120960}(36799,139849,-121797,123133,-88547,41499,-11351,1375)$.
7
1:
-11351/120960
7
2:
1537/4480
7
3:
-88547/120960
7
4:
123133/120960
7
5:
-4511/4480
7
6:
139849/120960
7
7:
5257/17280
8
0:
-33953/3628800
comment: 8-step Adams-Moulton formula, from newest to oldest: $\tfrac{1}{3628800}(1070017,4467094,-4604594,5595358,-5033120,3146338,-1291214,312874,-33953)$.
8
1:
156437/1814400
8
2:
-645607/1814400
8
3:
1573169/1814400
8
4:
-31457/22680
8
5:
2797679/1814400
8
6:
-2302297/1814400
8
7:
2233547/1814400
8
8:
1070017/3628800
9
0:
8183/1036800
comment: 9-step Adams-Moulton formula, from newest to oldest: $\tfrac{1}{7257600}(2082753,9449717,-11271304,16002320,-17283646,13510082,-7394032,2687864,-583435,57281)$.
9
1:
-116687/1451520
9
2:
335983/907200
9
3:
-462127/453600
9
4:
6755041/3628800
9
5:
-8641823/3628800
9
6:
200029/90720
9
7:
-1408913/907200
9
8:
9449717/7257600
9
9:
25713/89600
10
0:
-3250433/479001600
comment: 10-step Adams-Moulton formula, from newest to oldest: $\tfrac{1}{479001600}(134211265,656185652,-890175549,1446205080,-1823311566,1710774528,-1170597042,567450984,-184776195,36284876,-3250433)$.
10
1:
9071219/119750400
10
2:
-12318413/31933440
10
3:
23643791/19958400
10
4:
-21677723/8870400
10
5:
2227571/623700
10
6:
-33765029/8870400
10
7:
12051709/3991680
10
8:
-296725183/159667200
10
9:
164046413/119750400
10
10:
26842253/95800320
11
0:
4671/788480
comment: 11-step Adams-Moulton formula, from newest to oldest: $\tfrac{1}{958003200}(262747265,1374799219,-2092490673,3828828885,-5519460582,6043521486,-4963166514,3007739418,-1305971115,384709327,-68928781,5675265)$.
11
1:
-68928781/958003200
11
2:
384709327/958003200
11
3:
-87064741/63866880
11
4:
501289903/159667200
11
5:
-91910491/17740800
11
6:
1007253581/159667200
11
7:
-102212233/17740800
11
8:
36465037/9123840
11
9:
-99642413/45619200
11
10:
1374799219/958003200
11
11:
4777223/17418240
12
0:
-13695779093/2615348736000
comment: 12-step Adams-Moulton formula, from newest to oldest: $\tfrac{1}{2615348736000}(703604254357,3917551216986,-6616420957428,13465774256510,-21847538039895,27345870698436,-26204344465152,19058185652796,-10344711794985,4063327863170,-1092096992268,179842822566,-13695779093)$.
12
1:
2724891251/39626496000
12
2:
-30336027563/72648576000
12
3:
406332786317/261534873600
12
4:
-229882484333/58118860800
12
5:
529394045911/72648576000
12
6:
-4874320027/486486000
12
7:
84400835489/8072064000
12
8:
-485500845331/58118860800
12
9:
1346577425651/261534873600
12
10:
-551368413119/217945728000
12
11:
6595204069/4402944000
12
12:
703604254357/2615348736000
Definition
For $s\geq0$, the Adams–Moulton coefficients $\beta^*_{s,j}$ are the rational numbers for which the implicit $s$-step formula $y_{n+s}=y_{n+s-1}+h\sum_{j=0}^{s}\beta^*_{s,j}f_{n+j}$ is exact whenever $f(t,y(t))$ is a polynomial in $t$ of degree at most $s$ [1].
Parameters
$s$
—   number of steps ($s\geq0$)
$j$
—   index of the node, counted from the oldest value ($0\leq j\leq s$)
Formulas
(1)
If $\ell_{s,j}$ is the Lagrange basis polynomial for the nodes $0,1,\ldots,s$, with $\ell_{s,j}(i)=1$ for $i=j$ and $0$ otherwise, then $\beta^*_{s,j}=\int_{s-1}^{s}\ell_{s,j}(x)\,\mathrm{d}x$.
(2)
$\sum_{j=0}^{s}\beta^*_{s,j}j^m=(s^{m+1}-(s-1)^{m+1})/(m+1)$ for $0\leq m\leq s$.
(3)
$\beta^*_{s,0}=G_s=\psi_s(0)/s!$, where $G_s$ is the Gregory coefficient [4] and $\psi_s$ is the Bernoulli polynomial of the second kind.
Comments
(4)
Here $f_{n+j}=f(t_{n+j},y_{n+j})$ and $t_{n+j}=t_0+(n+j)h$. The case $s=0$ is backward Euler, and the case $s=1$ is the trapezoidal rule. The method has order $s+1$.
(5)
The step size appears as the single factor $h$ before the sum. The coefficient of $f_{n+j}$ in the formula is $h\beta^*_{s,j}$.
(6)
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 $(9,19,-5,1)/24$, while this table stores $j=0,1,2,3$ as $1/24,-5/24,19/24,3/8$.
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_moulton_coefficients(s):
    nodes = list(range(s + 1))
    rows = [[QQ(j) ** m for j in nodes] for m in range(s + 1)]
    rhs = [(QQ(s) ** (m + 1) - QQ(s - 1) ** (m + 1)) / QQ(m + 1)
           for m in range(s + 1)]
    return dict(zip(nodes, matrix(QQ, rows).solve_right(vector(QQ, rhs))))

adams_moulton_coefficients(3)
Links
Similar tables
Adams–Bashforth coefficients —   the explicit Adams predictor, using the previous $s$ values rather than including $f_{n+s}$
Lagrange basis polynomials for equally spaced nodes —   the coefficients are integrals of these basis polynomials over one unit step
Bernoulli polynomials of the second kind —   their value at $0$ gives the oldest Adams–Moulton coefficient: $\beta^*_{s,0}=\psi_s(0)/s!$
Butcher tableaux of explicit Runge–Kutta methods —   another family of rational coefficients for classical methods for ordinary differential equations
Data properties
Entries are of type: rational number
How well the digits are known: exact
Table is complete: no (it holds every coefficient with $0\leq s\leq12$, matching the Adams–Bashforth coefficient table so each predictor there has its Adams–Moulton corrector)
How they were obtained:

The generator builds the moment equations over $\mathbb{Q}$ for the nodes $0,1,\ldots,s$ and solves them exactly.

more

Each completed method is checked against the defining moments, the integral formula for the Lagrange basis polynomials, the displayed Adams–Moulton formulas through $s=4$ in [1], OEIS A260781 and A235936, and the Gregory-coefficient identity stated in the Formulas section.



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