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A032008 "AFK" (ordered, size, unlabeled) transform of 1,3,5,7,...

%I A032008 #17 Sep 11 2019 06:25:56
%S A032008 1,1,3,11,17,53,161,285,569,1459,4699,7177,15631,28229,66883,211311,
%T A032008 319929,627705,1163049,2150209,4422225,14320583,20392019,39962165,
%U A032008 68618087,126643545,212615483,433704811,1312156393,1864959757,3502343041,5919183485,10364053441
%N A032008 "AFK" (ordered, size, unlabeled) transform of 1,3,5,7,...
%C A032008 Number of compositions of n into distinct parts if there are (2i-1) kinds of part i. a(3) = 11: 3, 3', 3'', 3''', 3'''', 21, 2'1, 2''1, 12, 12', 12''. - _Alois P. Heinz_, Sep 05 2015
%H A032008 Alois P. Heinz, <a href="/A032008/b032008.txt">Table of n, a(n) for n = 0..4000</a>
%H A032008 C. G. Bower, <a href="/transforms2.html">Transforms (2)</a>
%p A032008 b:= proc(n, i, p) option remember;
%p A032008       `if`(n=0, p!, `if`(i<1, 0, b(n, i-1, p)+
%p A032008       `if`(i>n, 0, (2*i-1)*b(n-i, i-1, p+1))))
%p A032008     end:
%p A032008 a:= n-> b(n$2, 0):
%p A032008 seq(a(n), n=0..40);  # _Alois P. Heinz_, Sep 05 2015
%t A032008 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 i - 1)*b[n - i, i - 1, p + 1]]]];
%t A032008 a[n_] := b[n, n, 0];
%t A032008 a /@ Range[0, 40] (* _Jean-François Alcover_, Sep 11 2019, after _Alois P. Heinz_ *)
%o A032008 (PARI) seq(n)={apply(p->subst(serlaplace(p), y, 1), Vec(prod(k=1, n, 1 + (2*k-1)*x^k*y + O(x*x^n))))} \\ _Andrew Howroyd_, Jun 21 2018
%Y A032008 Cf. A032005, A032006, A032007.
%K A032008 nonn
%O A032008 0,3
%A A032008 _Christian G. Bower_
%E A032008 a(0)=1 prepended by _Alois P. Heinz_, Sep 05 2015