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

0,3

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)

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 *)

Prog

(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

Crossrefs

Cf. A022629.

Cf. A032005, A032006, A032008.

Sequence in context: A066967 A222450 A362683 * A091295 A361931 A215407

Adjacent sequences: A032004 A032005 A032006 * A032008 A032009 A032010

Keyword

nonn

Author

Christian G. Bower

Extensions

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

Status

approved