A330388 Expansion of e.g.f. Sum_{k>=1} (-1)^(k + 1) * log(1 + x)^k / (k * (1 - log(1 + x)^k)).

1, 0, 7, -37, 338, -2816, 28418, -340334, 5015080, -84244704, 1536606168, -29753884392, 609895549872, -13243687082016, 305507366834832, -7523621131117296, 198844500026698752, -5649686902983730560, 171839087043420258432, -5545292300345590210944

Offset

1,3

Links

Vaclav Kotesovec, Table of n, a(n) for n = 1..400

Formula

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

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

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

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

E.g.f.: Sum_{k>=1} log(1 + log(1 + x)^k). - Vaclav Kotesovec, Dec 15 2019

Conjecture: a(n) ~ n! * (-1)^(n+1) * Pi^2 * exp(n) / (24 * (exp(1) - 1)^(n+1)). - Vaclav Kotesovec, Dec 16 2019

Mathematica

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

Crossrefs

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

Sequence in context: A343568 A173766 A093168 * A371710 A276231 A097493

Adjacent sequences: A330385 A330386 A330387 * A330389 A330390 A330391

Keyword

sign

Author

Ilya Gutkovskiy, Dec 12 2019

Status

approved