A195287 - OEIS

%I #42 May 05 2019 03:24:23

%S 2,0,4,8,232,448,18224,35424,1036064,2025472,130960832,257072000,

%T 689908475264,1358275350528,8031885897472,15847920983552,

%U 7981032500085248,15774370258485248,12448755354530366464

%N a(n) = (A091137(n)/n!) * Integral_{u=-1..1} u*(u+1)*...*(u+n-1) du.

%C Numerators of the second row of an array based on Adams numerical integration. Take q!*s(m,q) = Integral_{-m-1..1} u*(u+1)*...*(u+q-1) du. a(n) is in the second row (case m=0) numerators of s(m,q) in the comments.

%C The unreduced array s(m,q), (m=-1,0,1,...,   columns q=0,1,2,...) is

%C 1,   1/2,   5/12,   9/24,   251/720, 475/1440,  = A002657(n)/A091137(n),

%C 2,     0,   4/12,   8/24,   232/720, 448/1440,  = a(n)/A091137(n),

%C 3,  -3/2,   9/12,   9/24,   243/720, 459/1440,

%C 4,  -8/2,  32/12,      0,   224/720, 448/1440,

%C 5, -15/2,  85/12, -55/24,   475/720, 475/1440,

%C 6, -24/2, 180/12, -216/24, 2376/720,        0.

%C Column numerators: A000027, -A067998(n), A152064(n), A157371(n), A165281(n).

%C Page 56 of the reference.

%C (*) 2/2 = 1,

%C     2/2 + 0 = 1,

%C     2/3 + 0 + 1/3 = 1,

%C     2/4 + 0 + 1/6 + 1/3 = 1. Reduced.

%D P. Curtz, Intégration numérique des systèmes differentiels à conditions initiales, Centre de Calcul Scientifique de l'Armement, Arcueil, 1969.

%F b(n) = a(n)/A091137(n).

%F b(0)/2 = 1,

%F b(0)/2 + b(1) = 1,

%F b(0)/3 + b(1)/2 + b(2) = 1,

%F b(0)/4 + b(1)/3 + b(2)/2 + b(3) = 1.

%F First vertical denominators: A028310(n) + 1. See A104661.

%F Values in (*).

%p A195287 := proc(n)

%p         mul(u+i,i=0..n-1) ;

%p         int(%,u=-1..1) ;

%p         %/n!*A091137(n) ;

%p end proc:

%p seq(A195287(n),n=0..20) ; # _R. J. Mathar_, Oct 02 2011

%t (* a7 = A091137 *) a7[n_] := a7[n] = Product[d, {d, Select[Divisors[n] + 1, PrimeQ]}]*a7[n-1]; a7[0]=1; a[n_] := a7[n]/n!*Integrate[ Pochhammer[u, n], {u, -1, 1}]; Table[a[n], {n, 0, 18}] (* _Jean-François Alcover_, Aug 13 2012 *)

%K nonn

%O 0,1

%A _Paul Curtz_, Sep 20 2011