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

%I A235936 #13 Jan 24 2014 19:35:27
%S A235936 1,1,1,5,8,-1,9,19,-5,1,251,646,-264,106,-19,475,1427,-798,482,-173,
%T A235936 27,19087,65112,-46461,37504,-20211,6312,-863,36799,139849,-121797,
%U A235936 123133,-88547,41499,-11351,1375
%N A235936 Triangle of numerators of the unreduced coefficients of a numerical integration for a prediction Adams method.
%C A235936 The coefficients b(q,j) are such that:
%C A235936 (q-j)!*j!*b(q,j) = (-1)^(q-j)*Int (from 0 to 1) u*(u-1)*...*(u-q) du/(u-j).
%C A235936 0<=j<=q, 0<=q<=p where p is the degree (or order) of the numerical integration.
%C A235936 This is the first case of tridimensional b(i,q,j), the integration is from i to i+1, with i=0.
%C A235936 The b(q,j) are:
%C A235936 1;
%C A235936 1/2,   1/2;
%C A235936 5/12, 8/12, -1/12;
%C A235936 9/24, 19/24, -5/24, 1/24;
%C A235936 ... etc.
%C A235936 The denominators are A232853(n).
%C A235936 The numerators are this sequence.
%C A235936 First column's numerators: A002657(n).
%C A235936 Main diagonal's numerators: (-1)^(n+1)*A141417(n).
%C A235936 Row sums are: 1,2,12,24,... (A091137).
%D A235936 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 A235936 Recurrence:
%F A235936 b(q,j) = (-1)^(q-j)*C(q,j)*b(q,q)+b(q-1,j).
%F A235936 C(q,j)=q!/((q-j)!*j!).
%e A235936 Triangle starts:
%e A235936 1;
%e A235936 1,     1;
%e A235936 5,     8,   -1;
%e A235936 9,    19,   -5,   1;
%e A235936 251, 646, -264, 106, -19;
%e A235936 ...
%e A235936 Numerators of
%e A235936 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 A235936 tabl,frac,sign
%O A235936 0,4
%A A235936 _Paul Curtz_, Jan 17 2014