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A300512 Expansion of e.g.f. log(Sum_{k>=0} p(k)*x^k/k!), where p(k) = number of partitions of k (A000041).

%I A300512 #14 Mar 12 2018 11:39:39
%S A300512 0,1,1,-1,-1,6,-1,-77,203,1344,-10692,-15862,579611,-1518768,
%T A300512 -32884753,283168220,1550435633,-38615194078,44538307279,
%U A300512 4920513118440,-39485852954288,-546206846420721,11322395643617278,23746787652752639,-2713550731461618505,17064642256532964421
%N A300512 Expansion of e.g.f. log(Sum_{k>=0} p(k)*x^k/k!), where p(k) = number of partitions of k (A000041).
%C A300512 Logarithmic transform of A000041.
%H A300512 Alois P. Heinz, <a href="/A300512/b300512.txt">Table of n, a(n) for n = 0..483</a>
%H A300512 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A300512 <a href="/index/Par#part">Index entries for related partition-counting sequences</a>
%F A300512 E.g.f.: log(Sum_{k>=0} A000041(k)*x^k/k!).
%e A300512 E.g.f.: A(x) = x/1! + x^2/2! - x^3/3! - x^4/4! + 6*x^5/5! - x^6/6! - 77*x^7/7! + 203*x^8/8! + ...
%p A300512 a:= proc(n) option remember; (t-> `if`(n=0, 0, t(n)-add(j*a(j)*
%p A300512       binomial(n, j)*t(n-j), j=1..n-1)/n))(combinat[numbpart])
%p A300512     end:
%p A300512 seq(a(n), n=0..30);  # _Alois P. Heinz_, Mar 07 2018
%t A300512 nmax = 25; CoefficientList[Series[Log[Sum[PartitionsP[k] x^k/k!, {k, 0, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
%t A300512 a[n_] := a[n] = PartitionsP[n] - Sum[k Binomial[n, k] PartitionsP[n - k] a[k], {k, 1, n - 1}]/n; a[0] = 0; Table[a[n], {n, 0, 25}]
%Y A300512 Cf. A000041, A038048, A300511.
%K A300512 sign
%O A300512 0,6
%A A300512 _Ilya Gutkovskiy_, Mar 07 2018