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

0,5

Links

Alois P. Heinz, Table of n, a(n) for n = 0..500

C. G. Bower, Transforms (2)

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

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 *)

Crossrefs

Cf. A003724.

Sequence in context: A316223 A087652 A072195 * A032220 A032145 A032050

Adjacent sequences: A032307 A032308 A032309 * A032311 A032312 A032313

Keyword

nonn

Author

Christian G. Bower

Extensions

Description corrected by Vladeta Jovovic, Aug 18 2004

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

Status

approved