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A307756 Exponential convolution of number of partitions into distinct parts (A000009) with themselves.

%I A307756 #11 May 06 2019 08:33:04
%S A307756 1,2,4,10,26,66,184,472,1268,3340,8748,22772,59102,151590,386830,
%T A307756 983914,2489384,6263284,15703204,39221884,97498736,241538472,
%U A307756 596115898,1465958522,3595196600,8788765304,21421616934,52080152238,126268822824,305365334180,736770528064
%N A307756 Exponential convolution of number of partitions into distinct parts (A000009) with themselves.
%H A307756 Vaclav Kotesovec, <a href="/A307756/b307756.txt">Table of n, a(n) for n = 0..3000</a>
%F A307756 E.g.f.: (Sum_{k>=0} A000009(k)*x^k/k!)^2.
%F A307756 a(n) = Sum_{k=0..n} binomial(n,k)*A000009(k)*A000009(n-k).
%F A307756 a(n) ~ exp(Pi*sqrt(2*n/3)) * 2^(n - 5/2) / (sqrt(3) * n^(3/2)). - _Vaclav Kotesovec_, May 06 2019
%p A307756 b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
%p A307756      `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
%p A307756     end:
%p A307756 a:= n-> add(binomial(n, j)*b(j)*b(n-j), j=0..n):
%p A307756 seq(a(n), n=0..30);  # _Alois P. Heinz_, Apr 26 2019
%t A307756 nmax = 30; CoefficientList[Series[Sum[PartitionsQ[k] x^k/k!, {k, 0, nmax}]^2, {x, 0, nmax}], x] Range[0, nmax]!
%t A307756 Table[Sum[Binomial[n, k] PartitionsQ[k] PartitionsQ[n - k], {k, 0, n}], {n, 0, 30}]
%Y A307756 Cf. A000009, A022567, A266232, A307755.
%K A307756 nonn
%O A307756 0,2
%A A307756 _Ilya Gutkovskiy_, Apr 26 2019