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 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 (list; table; graph; refs; listen; history; text; internal format)
 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. LINKS 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 Adjacent sequences:  A235933 A235934 A235935 * A235937 A235938 A235939 KEYWORD tabl,frac,sign AUTHOR Paul Curtz, Jan 17 2014 STATUS approved

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Last modified May 19 18:35 EDT 2022. Contains 353847 sequences. (Running on oeis4.)