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A130222 Number of partitions of 2n-set in which number of blocks of size 2k-1 is even (or zero) for every k.

%I A130222 #14 Nov 19 2020 07:51:55
%S A130222 1,2,11,117,2116,54233,1822053,76771684,3922196627,238355654605,
%T A130222 16936961517144,1387902030575371,129757092644981529,
%U A130222 13704639448111317852,1621528608322059614411,213338281602779271672663,31000779368619961156885708,4945841944762007645453032073
%N A130222 Number of partitions of 2n-set in which number of blocks of size 2k-1 is even (or zero) for every k.
%H A130222 Alois P. Heinz, <a href="/A130222/b130222.txt">Table of n, a(n) for n = 0..250</a>
%F A130222 E.g.f.: exp(cosh(x)-1)*Product_{k>0} cosh(x^(2*k-1)/(2*k-1)!).
%p A130222 A:= proc(n) exp(cosh(x)-1) *mul(cosh(x^(2*k-1)/ (2*k-1)!), k=1..n) end: a:= n-> coeff(series(A(n), x, 2*n+1), x, 2*n) *(2*n)!: seq(a(n), n=0..20); # _Alois P. Heinz_, Sep 29 2008
%p A130222 # second Maple program:
%p A130222 with(combinat):
%p A130222 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
%p A130222       `if`(irem(i, 2)=0 or j=0 or irem(j, 2)=0, multinomial(
%p A130222        n, n-i*j, i$j)/j!*b(n-i*j, i-1), 0), j=0..n/i)))
%p A130222     end:
%p A130222 a:= n-> b(2*n$2):
%p A130222 seq(a(n), n=0..30);  # _Alois P. Heinz_, Mar 08 2015
%t A130222 multinomial[n_, k_List] := n!/Times @@ (k!);
%t A130222 b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, Sum[If[Mod[i, 2]==0 || j==0 || Mod[j, 2]==0, multinomial[n, {n - i j} ~Join~ Table[i, {j}]]/j! b[n - i j, i-1], 0], {j, 0, n/i}]]];
%t A130222 a[n_] := b[2n, 2n];
%t A130222 a /@ Range[0, 30] (* _Jean-François Alcover_, Nov 19 2020, after _Alois P. Heinz_ *)
%Y A130222 Cf. A102759.
%K A130222 easy,nonn
%O A130222 0,2
%A A130222 _Vladeta Jovovic_, Aug 05 2007, Aug 05 2007
%E A130222 More terms from _Alois P. Heinz_, Sep 29 2008