A003709 E.g.f. cos(sin(x)) (even powers only). (Formerly M3986)

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

Offset

0,3

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

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

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 *)

Prog

(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 */

Crossrefs

Sequence in context: A318002 A323567 A304865 * A361281 A286928 A321042

Adjacent sequences: A003706 A003707 A003708 * A003710 A003711 A003712

Keyword

sign

Author

R. H. Hardin, Simon Plouffe

Status

approved