%I #13 Jan 24 2014 19:35:27
%S 1,1,1,5,8,-1,9,19,-5,1,251,646,-264,106,-19,475,1427,-798,482,-173,
%T 27,19087,65112,-46461,37504,-20211,6312,-863,36799,139849,-121797,
%U 123133,-88547,41499,-11351,1375
%N Triangle of numerators of the unreduced coefficients of a numerical integration for a prediction Adams method.
%C The coefficients b(q,j) are such that:
%C (q-j)!*j!*b(q,j) = (-1)^(q-j)*Int (from 0 to 1) u*(u-1)*...*(u-q) du/(u-j).
%C 0<=j<=q, 0<=q<=p where p is the degree (or order) of the numerical integration.
%C This is the first case of tridimensional b(i,q,j), the integration is from i to i+1, with i=0.
%C The b(q,j) are:
%C 1;
%C 1/2, 1/2;
%C 5/12, 8/12, -1/12;
%C 9/24, 19/24, -5/24, 1/24;
%C ... etc.
%C The denominators are A232853(n).
%C The numerators are this sequence.
%C First column's numerators: A002657(n).
%C Main diagonal's numerators: (-1)^(n+1)*A141417(n).
%C Row sums are: 1,2,12,24,... (A091137).
%D 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.
%F Recurrence:
%F b(q,j) = (-1)^(q-j)*C(q,j)*b(q,q)+b(q-1,j).
%F C(q,j)=q!/((q-j)!*j!).
%e Triangle starts:
%e 1;
%e 1, 1;
%e 5, 8, -1;
%e 9, 19, -5, 1;
%e 251, 646, -264, 106, -19;
%e ...
%e Numerators of
%e 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.
%K tabl,frac,sign
%O 0,4
%A _Paul Curtz_, Jan 17 2014
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