A032005 "AFK" (ordered, size, unlabeled) transform of 2,2,2,2,...

1, 2, 2, 10, 10, 18, 66, 74, 122, 178, 610, 666, 1146, 1586, 2450, 6778, 8026, 12738, 18258, 27194, 36938, 96226, 110578, 177930, 246474, 368354, 491426, 717418, 1543978, 1874418, 2855394, 3985322, 5765786, 7791250, 11066626, 14636538, 29870490, 35722514

Offset: 0

Author

Christian G. Bower

Keywords

• nonn

Comments

Number of compositions of n into distinct parts of 2 kinds. a(3) = 10: 3, 3', 21, 21', 2'1, 2'1', 12, 12', 1'2, 1'2'. - Alois P. Heinz, Sep 05 2015

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• C. G. Bower, Transforms (2)
• Jacob Sprittulla, On Colored Factorizations, arXiv:2008.09984 [math.CO], 2020.

Crossrefs

a(n) = 2 * A032043(n) - 2 for n>0.

Cf. A032006, A032007, 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, 2*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, 2*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 + 2*x^k*y + O(x*x^n))))} \\ Andrew Howroyd, Jun 21 2018

Extensions

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