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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A195287 a(n) = (A091137(n)/n!) * Integral_{u=-1..1} u*(u+1)*...*(u+n-1) du.

Original entry on oeis.org

2, 0, 4, 8, 232, 448, 18224, 35424, 1036064, 2025472, 130960832, 257072000, 689908475264, 1358275350528, 8031885897472, 15847920983552, 7981032500085248, 15774370258485248, 12448755354530366464
Offset: 0

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Author

Paul Curtz, Sep 20 2011

Keywords

Comments

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.
The unreduced array s(m,q), (m=-1,0,1,..., columns q=0,1,2,...) is
1, 1/2, 5/12, 9/24, 251/720, 475/1440, = A002657(n)/A091137(n),
2, 0, 4/12, 8/24, 232/720, 448/1440, = a(n)/A091137(n),
3, -3/2, 9/12, 9/24, 243/720, 459/1440,
4, -8/2, 32/12, 0, 224/720, 448/1440,
5, -15/2, 85/12, -55/24, 475/720, 475/1440,
6, -24/2, 180/12, -216/24, 2376/720, 0.
Column numerators: A000027, -A067998(n), A152064(n), A157371(n), A165281(n).
Page 56 of the reference.
(*) 2/2 = 1,
2/2 + 0 = 1,
2/3 + 0 + 1/3 = 1,
2/4 + 0 + 1/6 + 1/3 = 1. Reduced.

References

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

Programs

  • Maple
    A195287 := proc(n)
        mul(u+i,i=0..n-1) ;
        int(%,u=-1..1) ;
        %/n!*A091137(n) ;
end proc:
seq(A195287(n),n=0..20) ; # R. J. Mathar, Oct 02 2011
  • Mathematica
    (* 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 *)

Formula

b(n) = a(n)/A091137(n).
b(0)/2 = 1,
b(0)/2 + b(1) = 1,
b(0)/3 + b(1)/2 + b(2) = 1,
b(0)/4 + b(1)/3 + b(2)/2 + b(3) = 1.
First vertical denominators: A028310(n) + 1. See A104661.
Values in (*).
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