A330388 - OEIS

%I #12 Dec 16 2019 08:00:02

%S 1,0,7,-37,338,-2816,28418,-340334,5015080,-84244704,1536606168,

%T -29753884392,609895549872,-13243687082016,305507366834832,

%U -7523621131117296,198844500026698752,-5649686902983730560,171839087043420258432,-5545292300345590210944

%N Expansion of e.g.f. Sum_{k>=1} (-1)^(k + 1) * log(1 + x)^k / (k * (1 - log(1 + x)^k)).

%H Vaclav Kotesovec, <a href="/A330388/b330388.txt">Table of n, a(n) for n = 1..400</a>

%F E.g.f.: -Sum_{k>=1} log(1 - log(1 + x)^(2*k - 1)).

%F E.g.f.: A(x) = log(B(x)), where B(x) = e.g.f. of A298905.

%F exp(Sum_{n>=1} a(n) * (exp(x) - 1)^n / n!) = g.f. of A000009.

%F a(n) = Sum_{k=1..n} Stirling1(n,k) * (k - 1)! * A000593(k).

%F E.g.f.: Sum_{k>=1} log(1 + log(1 + x)^k). - _Vaclav Kotesovec_, Dec 15 2019

%F Conjecture: a(n) ~ n! * (-1)^(n+1) * Pi^2 * exp(n) / (24 * (exp(1) - 1)^(n+1)). - _Vaclav Kotesovec_, Dec 16 2019

%t nmax = 20; CoefficientList[Series[Sum[(-1)^(k + 1) Log[1 + x]^k/(k (1 - Log[1 + x]^k)), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest

%t Table[Sum[StirlingS1[n, k] (k - 1)! Sum[Mod[d, 2] d, {d, Divisors[k]}], {k, 1, n}], {n, 1, 20}]

%t nmax = 20; Rest[CoefficientList[Series[Sum[Log[1 + Log[1 + x]^k], {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!] (* _Vaclav Kotesovec_, Dec 15 2019 *)

%Y Cf. A000009, A000593, A008275, A088311, A089064, A265024, A298905, A330354, A330387.

%K sign

%O 1,3

%A _Ilya Gutkovskiy_, Dec 12 2019