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%I #23 Nov 05 2020 06:44:15
%S 1,2,2,10,10,18,66,74,122,178,610,666,1146,1586,2450,6778,8026,12738,
%T 18258,27194,36938,96226,110578,177930,246474,368354,491426,717418,
%U 1543978,1874418,2855394,3985322,5765786,7791250,11066626,14636538,29870490,35722514
%N "AFK" (ordered, size, unlabeled) transform of 2,2,2,2,...
%C 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
%H Alois P. Heinz, <a href="/A032005/b032005.txt">Table of n, a(n) for n = 0..1000</a>
%H C. G. Bower, <a href="/transforms2.html">Transforms (2)</a>
%H Jacob Sprittulla, <a href="https://arxiv.org/abs/2008.09984">On Colored Factorizations</a>, arXiv:2008.09984 [math.CO], 2020.
%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, 2*b(n-i, i-1, p+1))))
%p end:
%p a:= n-> b(n$2, 0):
%p seq(a(n), n=0..40); # _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, 2*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(prod(k=1, n, 1 + 2*x^k*y + O(x*x^n))))} \\ _Andrew Howroyd_, Jun 21 2018
%Y a(n) = 2 * A032043(n) - 2 for n>0.
%Y Cf. A032006, A032007, A032008.
%K nonn
%O 0,2
%A _Christian G. Bower_
%E a(0)=1 prepended by _Alois P. Heinz_, Sep 05 2015
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