A235936 Triangle of numerators of the unreduced coefficients of a numerical integration for a prediction Adams method.

1, 1, 1, 5, 8, -1, 9, 19, -5, 1, 251, 646, -264, 106, -19, 475, 1427, -798, 482, -173, 27, 19087, 65112, -46461, 37504, -20211, 6312, -863, 36799, 139849, -121797, 123133, -88547, 41499, -11351, 1375

Offset: 0

Author

Paul Curtz, Jan 17 2014

Keywords

• tabl
• frac
• sign

Comments

The coefficients b(q,j) are such that:

(q-j)!*j!*b(q,j) = (-1)^(q-j)*Int (from 0 to 1) u*(u-1)*...*(u-q) du/(u-j).

0<=j<=q, 0<=q<=p where p is the degree (or order) of the numerical integration.

This is the first case of tridimensional b(i,q,j), the integration is from i to i+1, with i=0.

The b(q,j) are:

1;

1/2, 1/2;

5/12, 8/12, -1/12;

9/24, 19/24, -5/24, 1/24;

... etc.

The denominators are A232853(n).

The numerators are this sequence.

First column's numerators: A002657(n).

Main diagonal's numerators: (-1)^(n+1)*A141417(n).

Row sums are: 1,2,12,24,... (A091137).

Examples

			Triangle starts:
1;
1,     1;
5,     8,   -1;
9,    19,   -5,   1;
251, 646, -264, 106, -19;
...
Numerators of
b(0,0)=1, b(1,0)=-(1/2-1)=1/2, b(1,1)=1/2, b(2,0)=(1/3-3/2+2)/2=5/12, b(2,1)=-(1/3-1)=2/3=8/12, b(2,2)=(1/3-1/2)/2=-1/12.
		

References

• P. Curtz, Intégration numérique des systèmes différentiels à conditions initiales, Note 12, Centre de Calcul Scientifique de l'Armement, Arcueil, (now DGA Maitrise de l'Information 35174 Bruz), 1969, see page 45.

Formula

Recurrence:

b(q,j) = (-1)^(q-j)*C(q,j)*b(q,q)+b(q-1,j).

C(q,j)=q!/((q-j)!*j!).