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

%I A032006 #22 Sep 11 2019 06:25:40
%S A032006 1,2,1,5,5,7,19,21,33,41,101,109,175,231,321,623,761,1087,1495,2109,
%T A032006 2661,4985,5849,8557,11251,15831,20373,27743,44357,55135,76123,101373,
%U A032006 136689,178673,240125,303997,475183,578271,793809,1024991,1387985,1763719,2363671
%N A032006 "AFK" (ordered, size, unlabeled) transform of 2,1,1,1,...
%C A032006 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 A032006 Alois P. Heinz, <a href="/A032006/b032006.txt">Table of n, a(n) for n = 0..1000</a>
%H A032006 C. G. Bower, <a href="/transforms2.html">Transforms (2)</a>
%p A032006 b:= proc(n, i, p) option remember;
%p A032006       `if`(n=0, p!, `if`(i<1, 0, b(n, i-1, p)+
%p A032006       `if`(i>n, 0, `if`(n=1, 2, 1)*b(n-i, i-1, p+1))))
%p A032006     end:
%p A032006 a:= n-> b(n$2, 0):
%p A032006 seq(a(n), n=0..45);  # _Alois P. Heinz_, Sep 05 2015
%t A032006 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 A032006 a[n_] := b[n, n, 0];
%t A032006 a /@ Range[0, 40] (* _Jean-François Alcover_, Sep 11 2019, after _Alois P. Heinz_ *)
%o A032006 (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 A032006 Cf. A032303.
%Y A032006 Cf. A032005, A032007, A032008.
%K A032006 nonn
%O A032006 0,2
%A A032006 _Christian G. Bower_
%E A032006 a(0)=1 prepended by _Alois P. Heinz_, Sep 05 2015