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%I #16 Mar 16 2015 09:09:24
%S 1,1,1,2,8,27,82,338,1647,7668,37779,210520,1276662,7985200,51302500,
%T 358798144,2677814900,20309850311,160547934756,1344197852830,
%U 11666610870142,104156661915427,962681713955130,9238216839975106,91508384728188792,930538977116673878
%N Number of partitions of n-set in which number of blocks of size 2k is even (or zero) for every k.
%H Alois P. Heinz, <a href="/A102759/b102759.txt">Table of n, a(n) for n = 0..500</a>
%F E.g.f. for offset 2: exp(sinh(x))*Product_{k>=1} cosh(x^(2*k)/(2*k)!). - _Geoffrey Critzer_, Jan 02 2011
%p with(combinat):
%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p add(`if`(irem(i, 2)=1 or irem(j, 2)=0, multinomial(
%p n, n-i*j, i$j)/j!*b(n-i*j, i-1), 0), j=0..n/i)))
%p end:
%p a:= n-> b(n$2):
%p seq(a(n), n=0..30); # _Alois P. Heinz_, Mar 08 2015
%t multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[If[Mod[i, 2] == 1 || Mod[j, 2] == 0, multinomial[n, Join[{n-i*j}, Table[i, {j}]]]/j!*b[n-i*j, i-1], 0], {j, 0, n/i}]]] ; a[n_] := b[n, n]; Table[ a[n], {n, 0, 30}] (* _Jean-François Alcover_, Mar 16 2015, after _Alois P. Heinz_ *)
%o (PARI) N=31; x='x+O('x^N);
%o Vec(serlaplace(exp(sinh(x))*prod(k=1,N,cosh(x^(2*k)/(2*k)!))))
%o /* gives: [1, 1, 1, 2, 8, 27, 82, 338, 1647, 7668, ...] , _Joerg Arndt_, Jan 03 2011 */
%Y Cf. A003483, A006950, A130219 - A130223.
%K nonn
%O 0,4
%A _Vladeta Jovovic_, Feb 10 2005, Aug 05 2007
%E Offset changed to 0 and two 1's prepended by _Alois P. Heinz_, Mar 08 2015
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