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A032007 "AFK" (ordered, size, unlabeled) transform of 1,2,3,4,...

%I A032007 #22 Sep 11 2019 06:25:48
%S A032007 1,1,2,7,10,25,68,111,208,435,1218,1773,3586,6077,12156,31961,47624,
%T A032007 86825,151962,265525,469610,1242607,1750108,3217663,5263928,9205197,
%U A032007 14713474,26440503,63610938,90877893,159360628,258871127,431309688,687140639,1134231986
%N A032007 "AFK" (ordered, size, unlabeled) transform of 1,2,3,4,...
%C A032007 Sum of products of parts in all compositions of n into distinct parts. - _Vladeta Jovovic_, Feb 21 2005
%C A032007 Number of compositions of n into distinct parts if there are i kinds of part i. a(3) = 7: 3, 3', 3'', 21, 2'1, 12, 12'.
%H A032007 Alois P. Heinz, <a href="/A032007/b032007.txt">Table of n, a(n) for n = 0..5000</a>
%H A032007 C. G. Bower, <a href="/transforms2.html">Transforms (2)</a>
%p A032007 b:= proc(n, i, p) option remember;
%p A032007       `if`(n=0, p!, `if`(i<1, 0, b(n, i-1, p)+
%p A032007       `if`(i>n, 0, i*b(n-i, i-1, p+1))))
%p A032007     end:
%p A032007 a:= n-> b(n$2, 0):
%p A032007 seq(a(n), n=0..40);  # _Alois P. Heinz_, Sep 05 2015
%t A032007 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, i*b[n - i, i - 1, p + 1]]]];
%t A032007 a[n_] := b[n, n, 0];
%t A032007 a /@ Range[0, 40] (* _Jean-François Alcover_, Sep 11 2019, after _Alois P. Heinz_ *)
%o A032007 (PARI) seq(n)={apply(p->subst(serlaplace(p), y, 1), Vec(prod(k=1, n, 1 + k*x^k*y + O(x*x^n))))} \\ _Andrew Howroyd_, Jun 21 2018
%Y A032007 Cf. A022629.
%Y A032007 Cf. A032005, A032006, A032008.
%K A032007 nonn
%O A032007 0,3
%A A032007 _Christian G. Bower_
%E A032007 a(0)=1 prepended by _Alois P. Heinz_, Sep 05 2015