login
A235936
Triangle of numerators of the unreduced coefficients of a numerical integration for a prediction Adams method.
0
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
0,4
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).
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!).
EXAMPLE
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.
CROSSREFS
Sequence in context: A185393 A073333 A316229 * A260781 A306068 A021949
KEYWORD
tabl,frac,sign
AUTHOR
Paul Curtz, Jan 17 2014
STATUS
approved