A003709 E.g.f. cos(sin(x)) (even powers only).

1, -1, 5, -37, 457, -8169, 188685, -5497741, 197920145, -8541537105, 432381471509, -25340238127989, 1699894200469849, -129076687233903673, 10989863562589199389, -1041327644107761435101, 109095160722852951673633, -12561989444137938396142753

Offset: 0

Author

R. H. Hardin, Simon Plouffe

Keywords

• sign

Comments

|a(n)| is the number of ways to partition the set {1,2,...,2n} into an even number of odd size blocks. - Geoffrey Critzer, Apr 11 2010

Unsigned sequence has e.g.f. cosh(sinh(x)) (even powers only).

References

• L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 226, 8th line of table.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 0..50

Programs

Maple

b:= proc(n) option remember; `if`(n=0, 1, add(
      b(n-j)*irem(j, 2)*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> b(2*n)*(-1)^n:
seq(a(n), n=0..20);  # Alois P. Heinz, Feb 11 2023

Mathematica

Take[With[{nn=40},CoefficientList[Series[Cos[Sin[x]],{x,0,nn}],x] Range[0,nn]!],{1,-1,2}] (* Harvey P. Dale, Sep 18 2011 *)

Maxima

a(n):=sum((2^(2*j+1)*sum((i-n+j)^(2*n)*binomial((2*n-2*j),i)*(-1)^(n-i),i,0,(n-j))/(2*n-2*j)!),j,0,n); /* Vladimir Kruchinin, Jun 08 2011 */

Formula

a(n) = sum(j=0..n, (2^(2*j+1)*sum(i=0..(n-j), (i-n+j)^(2*n)*binomial((2*n-2*j),i)*(-1)^(n-i))/(2*n-2*j)!)), n>0, a(1)=0. - Vladimir Kruchinin, Jun 08 2011