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A300514 Expansion of e.g.f. exp(Sum_{k>=1} q(k)*x^k/k!), where q(k) = number of partitions of k into distinct parts (A000009).

%I A300514 #9 Mar 11 2018 19:47:23
%S A300514 1,1,2,6,20,79,358,1791,9854,58958,379716,2617320,19197327,149099827,
%T A300514 1221390172,10515829901,94865603724,894302028718,8788782784778,
%U A300514 89848652800152,953666248076772,10491219933196228,119429574273909421,1404835599743325765,17052591331677804136
%N A300514 Expansion of e.g.f. exp(Sum_{k>=1} q(k)*x^k/k!), where q(k) = number of partitions of k into distinct parts (A000009).
%C A300514 Exponential transform of A000009.
%H A300514 Alois P. Heinz, <a href="/A300514/b300514.txt">Table of n, a(n) for n = 0..550</a>
%H A300514 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A300514 <a href="/index/Par#part">Index entries for related partition-counting sequences</a>
%F A300514 E.g.f.: exp(Sum_{k>=1} A000009(k)*x^k/k!).
%e A300514 E.g.f.: A(x) = 1 + x/1! + 2*x^2/2! + 6*x^3/3! + 20*x^4/4! + 79*x^5/5! + 358*x^6/6! + 1791*x^7/7! + ...
%p A300514 b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
%p A300514      `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
%p A300514     end:
%p A300514 a:= proc(n) option remember; `if`(n=0, 1, add(
%p A300514       a(n-j)*binomial(n-1, j-1)*b(j), j=1..n))
%p A300514     end:
%p A300514 seq(a(n), n=0..30);  # _Alois P. Heinz_, Mar 07 2018
%t A300514 nmax = 24; CoefficientList[Series[Exp[Sum[PartitionsQ[k] x^k/k!, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
%t A300514 a[n_] := a[n] = Sum[PartitionsQ[k] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 24}]
%Y A300514 Cf. A000009, A293839, A293840, A300511, A300515.
%K A300514 nonn
%O A300514 0,3
%A A300514 _Ilya Gutkovskiy_, Mar 07 2018