cp's OEIS Frontend

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

%I A003709 M3986 #30 Feb 11 2023 12:21:59
%S A003709 1,-1,5,-37,457,-8169,188685,-5497741,197920145,-8541537105,
%T A003709 432381471509,-25340238127989,1699894200469849,-129076687233903673,
%U A003709 10989863562589199389,-1041327644107761435101,109095160722852951673633,-12561989444137938396142753
%N A003709 E.g.f. cos(sin(x)) (even powers only).
%C A003709 |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
%C A003709 Unsigned sequence has e.g.f. cosh(sinh(x)) (even powers only).
%D A003709 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 226, 8th line of table.
%D A003709 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A003709 T. D. Noe, <a href="/A003709/b003709.txt">Table of n, a(n) for n = 0..50</a>
%F A003709 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
%p A003709 b:= proc(n) option remember; `if`(n=0, 1, add(
%p A003709       b(n-j)*irem(j, 2)*binomial(n-1, j-1), j=1..n))
%p A003709     end:
%p A003709 a:= n-> b(2*n)*(-1)^n:
%p A003709 seq(a(n), n=0..20);  # _Alois P. Heinz_, Feb 11 2023
%t A003709 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 *)
%o A003709 (Maxima)
%o A003709 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 */
%K A003709 sign
%O A003709 0,3
%A A003709 _R. H. Hardin_, _Simon Plouffe_