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A324237 Expansion of e.g.f. 1 / (1 - Sum_{k>=1} q(k)*x^k/k!), where q(k) = number of partitions of k into distinct parts (A000009).

%I A324237 #21 Sep 03 2019 03:54:55
%S A324237 1,1,3,14,84,633,5730,60485,729710,9904064,149358998,2477662364,
%T A324237 44837516675,879028693860,18558771941586,419815668642109,
%U A324237 10129704474860688,259695154154923814,7049438079064414206,201988316828399901634,6092203404985463075656
%N A324237 Expansion of e.g.f. 1 / (1 - Sum_{k>=1} q(k)*x^k/k!), where q(k) = number of partitions of k into distinct parts (A000009).
%H A324237 Vaclav Kotesovec, <a href="/A324237/b324237.txt">Table of n, a(n) for n = 0..419</a>
%F A324237 a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * q(k) * a(n-k).
%F A324237 a(n) ~ c * d^n * n!, where d = 1.5080583621492799630678624980320180394686208919872154400104169910221003637... and c = 0.67652958824662835367141799671720225317465169475061770258661897351... - _Vaclav Kotesovec_, Sep 03 2019
%p A324237 b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
%p A324237      `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
%p A324237     end:
%p A324237 a:= proc(n) option remember; `if`(n=0, 1, add(
%p A324237       a(n-k)*binomial(n, k)*b(k), k=1..n))
%p A324237     end:
%p A324237 seq(a(n), n=0..20);  # _Alois P. Heinz_, Sep 02 2019
%t A324237 nmax = 21; CoefficientList[Series[1/(1 - Sum[PartitionsQ[k] x^k/k!, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!
%t A324237 a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] PartitionsQ[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 21}]
%Y A324237 Cf. A000009, A266232, A300514, A304969, A324236.
%K A324237 nonn
%O A324237 0,3
%A A324237 _Ilya Gutkovskiy_, Sep 02 2019