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

1, 2, 11, 117, 2116, 54233, 1822053, 76771684, 3922196627, 238355654605, 16936961517144, 1387902030575371, 129757092644981529, 13704639448111317852, 1621528608322059614411, 213338281602779271672663, 31000779368619961156885708, 4945841944762007645453032073

Offset: 0

Author

Vladeta Jovovic, Aug 05 2007, Aug 05 2007

Keywords

• easy
• nonn

Links

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

Crossrefs

Cf. A102759.

Programs

Maple

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
# second Maple program:
with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      `if`(irem(i, 2)=0 or j=0 or irem(j, 2)=0, multinomial(
       n, n-i*j, i$j)/j!*b(n-i*j, i-1), 0), j=0..n/i)))
    end:
a:= n-> b(2*n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 08 2015

Mathematica

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]==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}]]];
a[n_] := b[2n, 2n];
a /@ Range[0, 30] (* Jean-François Alcover, Nov 19 2020, after Alois P. Heinz *)

Formula

E.g.f.: exp(cosh(x)-1)*Product_{k>0} cosh(x^(2*k-1)/(2*k-1)!).

Extensions

More terms from Alois P. Heinz, Sep 29 2008