A032006 "AFK" (ordered, size, unlabeled) transform of 2,1,1,1,...

1, 2, 1, 5, 5, 7, 19, 21, 33, 41, 101, 109, 175, 231, 321, 623, 761, 1087, 1495, 2109, 2661, 4985, 5849, 8557, 11251, 15831, 20373, 27743, 44357, 55135, 76123, 101373, 136689, 178673, 240125, 303997, 475183, 578271, 793809, 1024991, 1387985, 1763719, 2363671

Offset

0,2

Comments

Number of compositions of n into distinct parts if there are 2 kinds of part 1. a(3) = 5: 3, 21, 21', 12, 1'2.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

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, `if`(n=1, 2, 1)*b(n-i, i-1, p+1))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..45);  # 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, If[n == 1, 2, 1]*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((1 + 2*x*y)*prod(k=2, n, 1 + x^k*y + O(x*x^n))))} \\ Andrew Howroyd, Jun 21 2018

Crossrefs

Cf. A032303.

Cf. A032005, A032007, A032008.

Sequence in context: A193536 A152290 A248699 * A167158 A074392 A284428

Adjacent sequences: A032003 A032004 A032005 * A032007 A032008 A032009

Keyword

nonn

Author

Christian G. Bower

Extensions

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

Status

approved