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A130221 Number of partitions of n-set in which number of blocks of size 2k is odd (or zero) for every k.

%I A130221 #13 Mar 24 2019 22:44:34
%S A130221 1,1,2,5,12,37,158,667,2740,13461,74710,412095,2406880,15450541,
%T A130221 103187698,715323395,5236160612,40014337437,318488475658,
%U A130221 2637143123027,22603231117364,201268520010153,1855401760331982,17624602999352535,173071602624629536
%N A130221 Number of partitions of n-set in which number of blocks of size 2k is odd (or zero) for every k.
%H A130221 Alois P. Heinz, <a href="/A130221/b130221.txt">Table of n, a(n) for n = 0..500</a>
%F A130221 E.g.f.: exp(sinh(x))*Product_{k>0} (1+sinh(x^(2*k)/(2*k)!)).
%e A130221 a(4)=12 because from the 15 (=A000110(4)) partitions of the 4-set {a,b,c,d} only the partitions ab|cd, ac|bd and ad|bc do not qualify.
%p A130221 g:=exp(sinh(x))*(product(1+sinh(x^(2*k)/factorial(2*k)), k=1..25)): gser:= series(g,x=0,30): seq(factorial(n)*coeff(gser,x,n),n=0..23); # _Emeric Deutsch_, Aug 28 2007
%p A130221 # second Maple program:
%p A130221 with(combinat):
%p A130221 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
%p A130221        `if`(j=0 or irem(i, 2)=1 or irem(j, 2)=1, multinomial(
%p A130221         n, n-i*j, i$j)/j!*b(n-i*j, i-1), 0), j=0..n/i)))
%p A130221     end:
%p A130221 a:= n-> b(n$2):
%p A130221 seq(a(n), n=0..30);  # _Alois P. Heinz_, Mar 08 2015
%t A130221 multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[If[j == 0 || Mod[i, 2] == 1 || Mod[j, 2] == 1, multinomial[n, Join[{ n - i*j}, Array[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_, Dec 22 2016, after _Alois P. Heinz_ *)
%Y A130221 Cf. A000110, A102759.
%K A130221 easy,nonn
%O A130221 0,3
%A A130221 _Vladeta Jovovic_, Aug 05 2007, Aug 05 2007
%E A130221 More terms from _Emeric Deutsch_, Aug 28 2007