A032006 - OEIS

%I #22 Sep 11 2019 06:25:40

%S 1,2,1,5,5,7,19,21,33,41,101,109,175,231,321,623,761,1087,1495,2109,

%T 2661,4985,5849,8557,11251,15831,20373,27743,44357,55135,76123,101373,

%U 136689,178673,240125,303997,475183,578271,793809,1024991,1387985,1763719,2363671

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

%C 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.

%H Alois P. Heinz, <a href="/A032006/b032006.txt">Table of n, a(n) for n = 0..1000</a>

%H C. G. Bower, <a href="/transforms2.html">Transforms (2)</a>

%p b:= proc(n, i, p) option remember;

%p       `if`(n=0, p!, `if`(i<1, 0, b(n, i-1, p)+

%p       `if`(i>n, 0, `if`(n=1, 2, 1)*b(n-i, i-1, p+1))))

%p     end:

%p a:= n-> b(n$2, 0):

%p seq(a(n), n=0..45);  # _Alois P. Heinz_, Sep 05 2015

%t 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]]]];

%t a[n_] := b[n, n, 0];

%t a /@ Range[0, 40] (* _Jean-François Alcover_, Sep 11 2019, after _Alois P. Heinz_ *)

%o (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

%Y Cf. A032303.

%Y Cf. A032005, A032007, A032008.

%K nonn

%O 0,2

%A _Christian G. Bower_

%E a(0)=1 prepended by _Alois P. Heinz_, Sep 05 2015