A130219 Number of partitions of 2n-set in which number of blocks of size k is even (or zero) for every k.

1, 1, 4, 56, 631, 15457, 582374, 18589286, 894499204, 51154344582, 3823359163826, 274722100927166, 25458967562911128, 2569179797929092506, 284554990016509385086, 37830153187190688287522, 5093072752898942262610007, 798814778335473578083666573

Offset: 0

Author

Vladeta Jovovic, Aug 04 2007, Aug 05 2007

Keywords

• easy
• nonn

Examples

			a(2)=4 because we have ab|cd, ac|bd, ad|bc and a|b|c|d.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..250

Crossrefs

Cf. A055922, A102759, A111723, A111724.

Programs

Maple

g:=product(cosh(x^k/factorial(k)),k=1..35): gser:=series(g,x=0,32): seq(factorial(2*n)*coeff(gser,x,2*n),n=0..14); # Emeric Deutsch, Sep 01 2007
# second Maple program:
g:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<1, 0, add(
     `if`(irem(j, 2)=0, g(n-i*j, i-1, p+j*i)/j!/i!^j, 0), j=0..n/i)))
    end:
a:= n-> g(2*n$2, 0):
seq(a(n), n=0..20);  # Alois P. Heinz, Mar 06 2015

Mathematica

g[n_, i_, p_] := g[n, i, p] = If[n == 0, p!, If[i<1, 0, Sum[If[Mod[j, 2] == 0, g[n - i*j, i-1, p + j*i]/j!/i!^j, 0], {j, 0, n/i}]]]; a[n_] := g[2*n, 2*n, 0]; Table[ a[n], {n, 0, 20}] (* Jean-François Alcover, May 12 2015, after Alois P. Heinz *)

Formula

E.g.f.: Product_{k>0} cosh(x^k/k!).

Extensions

More terms from Emeric Deutsch, Sep 01 2007