A032013 Number of ways to partition n labeled elements into sets of different sizes of at least 2 and order the sets.

1, 0, 1, 1, 1, 21, 31, 113, 169, 8053, 15871, 71325, 300147, 816401, 63105953, 161203747, 856049593, 4050514725, 25570388671, 80377109117, 12126315199099, 36747628912981, 233849676829957, 1239662165799711, 8321234529548651, 59953576690379081

Offset: 0

Author

Christian G. Bower

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..673
• C. G. Bower, Transforms (2)

Crossrefs

Cf. A032011.

Programs

Maple

b:= proc(n, i, p) option remember;
      `if`(n=0, p!, `if`(i<2, 0, b(n, i-1, p)+
      `if`(i>n, 0, b(n-i, i-1, p+1)*binomial(n, i))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..30);  # Alois P. Heinz, May 11 2016

Mathematica

b[n_, i_, p_] := b[n, i, p] = If[n == 0, p!, If[i < 2, 0, b[n, i - 1, p] + If[i > n, 0, b[n - i, i - 1, p + 1]*Binomial[n, i]]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 27 2017, after Alois P. Heinz *)

PARI

seq(n)=[subst(serlaplace(y^0*p),y,1) | p <- Vec(serlaplace(prod(k=2, n, 1 + x^k*y/k! + O(x*x^n))))] \\ Andrew Howroyd, Sep 13 2018

Formula

"AGJ" (ordered, elements, labeled) transform of 0, 1, 1, 1...

Extensions

a(0)=1 prepended by Alois P. Heinz, May 11 2016