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

0,2

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.

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 *)

Prog

(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

Crossrefs

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

Cf. A032006, A032007, A032008.

Sequence in context: A309751 A249152 A216708 * A147801 A367545 A263053

Adjacent sequences: A032002 A032003 A032004 * A032006 A032007 A032008

Keyword

nonn

Author

Christian G. Bower

Extensions

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

Status

approved