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A300515 Expansion of e.g.f. log(Sum_{k>=0} q(k)*x^k/k!), where q(k) = number of partitions of k into distinct parts (A000009).

%I A300515 #8 Mar 11 2018 19:49:48
%S A300515 0,1,0,1,-3,7,-24,130,-748,4446,-30694,245586,-2131621,19850237,
%T A300515 -201363613,2214638141,-26037523804,325653856386,-4331545709166,
%U A300515 61069238694738,-908488414975896,14220161323121232,-233746798117055047,4025924893291859919,-72487584601341680720
%N A300515 Expansion of e.g.f. log(Sum_{k>=0} q(k)*x^k/k!), where q(k) = number of partitions of k into distinct parts (A000009).
%C A300515 Logarithmic transform of A000009.
%H A300515 Alois P. Heinz, <a href="/A300515/b300515.txt">Table of n, a(n) for n = 0..470</a>
%H A300515 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A300515 <a href="/index/Par#part">Index entries for related partition-counting sequences</a>
%F A300515 E.g.f.: log(Sum_{k>=0} A000009(k)*x^k/k!).
%e A300515 E.g.f.: A(x) = x/1! + x^3/3! - 3*x^4/4! + 7*x^5/5! - 24*x^6/6! + 130*x^7/7! - 748*x^8/8! + ...
%p A300515 b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
%p A300515      `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
%p A300515     end:
%p A300515 a:= proc(n) option remember; (t-> `if`(n=0, 0, t(n)-add(
%p A300515       j*a(j)*binomial(n, j)*t(n-j), j=1..n-1)/n))(b)
%p A300515     end:
%p A300515 seq(a(n), n=0..30);  # _Alois P. Heinz_, Mar 07 2018
%t A300515 nmax = 24; CoefficientList[Series[Log[Sum[PartitionsQ[k] x^k/k!, {k, 0, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
%t A300515 a[n_] := a[n] = PartitionsQ[n] - Sum[k Binomial[n, k] PartitionsQ[n - k] a[k], {k, 1, n - 1}]/n; a[0] = 0; Table[a[n], {n, 0, 24}]
%Y A300515 Cf. A000009, A265024, A300512, A300514.
%K A300515 sign
%O A300515 0,5
%A A300515 _Ilya Gutkovskiy_, Mar 07 2018