A032007 "AFK" (ordered, size, unlabeled) transform of 1,2,3,4,...

1, 1, 2, 7, 10, 25, 68, 111, 208, 435, 1218, 1773, 3586, 6077, 12156, 31961, 47624, 86825, 151962, 265525, 469610, 1242607, 1750108, 3217663, 5263928, 9205197, 14713474, 26440503, 63610938, 90877893, 159360628, 258871127, 431309688, 687140639, 1134231986

Offset: 0

Author

Christian G. Bower

Keywords

• nonn

Comments

Sum of products of parts in all compositions of n into distinct parts. - Vladeta Jovovic, Feb 21 2005

Number of compositions of n into distinct parts if there are i kinds of part i. a(3) = 7: 3, 3', 3'', 21, 2'1, 12, 12'.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..5000
• C. G. Bower, Transforms (2)

Crossrefs

Cf. A022629.

Cf. A032005, A032006, A032008.

Programs

Maple

b:= proc(n, i, p) option remember;
      `if`(n=0, p!, `if`(i<1, 0, b(n, i-1, p)+
      `if`(i>n, 0, i*b(n-i, i-1, p+1))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..40);  # Alois P. Heinz, Sep 05 2015

Mathematica

b[n_, i_, p_] := b[n, i, p] = If[n == 0, p!, If[i < 1, 0, b[n, i - 1, p] + If[i > n, 0, i*b[n - i, i - 1, p + 1]]]];
a[n_] := b[n, n, 0];
a /@ Range[0, 40] (* Jean-François Alcover, Sep 11 2019, after Alois P. Heinz *)

PARI

seq(n)={apply(p->subst(serlaplace(p), y, 1), Vec(prod(k=1, n, 1 + k*x^k*y + O(x*x^n))))} \\ Andrew Howroyd, Jun 21 2018

Extensions

a(0)=1 prepended by Alois P. Heinz, Sep 05 2015