A330387 Expansion of e.g.f. Sum_{k>=1} (-1)^(k + 1) * (exp(x) - 1)^k / (k * (1 - (exp(x) - 1)^k)).

1, 2, 12, 62, 420, 3782, 40572, 463262, 5708820, 80773622, 1319927532, 23675250062, 447145154820, 8830952572262, 185694817024092, 4246473212654462, 105754322266866420, 2811068529133151702, 78039884046777282252, 2243558766132057764462

Offset

1,2

Links

Table of n, a(n) for n=1..20.

Formula

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

E.g.f.: A(x) = log(B(x)), where B(x) = e.g.f. of A305550.

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

a(n) = Sum_{k=1..n} Stirling2(n,k) * (k - 1)! * A000593(k).

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

a(n) ~ n! * Pi^2 / (24 * (log(2))^(n+1)). - Vaclav Kotesovec, Dec 15 2019

Mathematica

nmax = 20; CoefficientList[Series[Sum[(-1)^(k + 1) (Exp[x] - 1)^k/(k (1 - (Exp[x] - 1)^k)), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
Table[Sum[StirlingS2[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 + (Exp[x] - 1)^k], {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!] (* Vaclav Kotesovec, Dec 15 2019 *)

Crossrefs

Cf. A000009, A000593, A000629, A008277, A088311, A265024, A305550, A330353, A330388.

Sequence in context: A026076 A361812 A231025 * A368760 A272363 A283488

Adjacent sequences: A330384 A330385 A330386 * A330388 A330389 A330390

Keyword

nonn

Author

Ilya Gutkovskiy, Dec 12 2019

Status

approved