A032310 Number of ways to partition n labeled elements into sets of odd sizes, with all sizes different.

1, 1, 0, 1, 4, 1, 6, 1, 64, 505, 130, 1321, 1024, 13157, 2380, 395851, 5782144, 1639617, 24545706, 16100905, 306621184, 292018525, 6304002100, 1549052715, 507969498304, 11794047630801, 3164830777316, 75389026652551, 48756350408224, 1240389053007865

Offset: 0

Author

Christian G. Bower

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..500
• C. G. Bower, Transforms (2)

Crossrefs

Cf. A003724.

Programs

Maple

with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-2), j=0..min(1, n/i))))
    end:
a:= n-> b(n, iquo(n+1, 2)*2-1):
seq(a(n), n=0..40);  # 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[multinomial[n, Join[{n-i*j}, Array[i&, j]]]/j!*b[n - i*j, i-2], {j, 0, Min[1, n/i]}]]]; a[n_] := b[n, Quotient[n+1, 2]*2-1]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 05 2017, after Alois P. Heinz *)

Formula

"EGJ" (unordered, element, labeled) transform of 1, 0, 1, 0... (odds)

E.g.f.: Product_{k>0} (1+x^(2*k-1)/(2*k-1)!). - Vladeta Jovovic, Jan 16 2004

Extensions

Description corrected by Vladeta Jovovic, Aug 18 2004

a(0)=1 prepended by Alois P. Heinz, Mar 08 2015