A308555 -id:A308555 - cp's OEIS frontend

A330353 Expansion of e.g.f. Sum_{k>=1} (exp(x) - 1)^k / (k * (1 - (exp(x) - 1)^k)).

1, 4, 18, 112, 810, 7144, 73458, 850672, 11069370, 161190904, 2575237698, 44571447232, 836188737930, 16970931765064, 368985732635538, 8524290269083792, 208874053200038490, 5428866923032585624, 149250273758730282978, 4318265042184721248352

Offset: 1

Ilya Gutkovskiy, Dec 11 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..420

Cf. A000041, A000203, A000629, A002745, A008277, A038048, A167137, A308555, A330351, A330352, A330354.

nmax = 20; CoefficientList[Series[Sum[(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)! DivisorSigma[1, k], {k, 1, n}], {n, 1, 20}]

E.g.f.: -Sum_{k>=1} log(1 - (exp(x) - 1)^k).
E.g.f.: A(x) = log(B(x)), where B(x) = e.g.f. of A167137.
G.f.: Sum_{k>=1} (k - 1)! * sigma(k) * x^k / Product_{j=1..k} (1 - j*x), where sigma = A000203.
exp(Sum_{n>=1} a(n) * log(1 + x)^n / n!) = g.f. of the partition numbers (A000041).
a(n) = Sum_{k=1..n} Stirling2(n,k) * (k - 1)! * sigma(k).
a(n) ~ n! * Pi^2 / (12 * (log(2))^(n+1)). - Vaclav Kotesovec, Dec 14 2019

A308554 Expansion of e.g.f. Sum_{k>=1} tau(k)*(exp(x) - 1)^k/k!, where tau = number of divisors (A000005).

Original entry on oeis.org

1, 3, 9, 30, 113, 472, 2145, 10514, 55428, 313255, 1886888, 12029741, 80701715, 567541878, 4175795147, 32104799401, 257561662496, 2151841672173, 18676002357864, 167951667633495, 1561420657033927, 14980472336450530, 148140814019762129, 1508776236781766431

Offset: 1

Ilya Gutkovskiy, Jun 07 2019

nonn

Stirling transform of A000005.

Alois P. Heinz, Table of n, a(n) for n = 1..575

Cf. A000005, A160399, A308555.

b:= proc(n, m) option remember; uses numtheory;
     `if`(n=0, tau(m), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=1..24);  # Alois P. Heinz, Aug 04 2021

nmax = 24; Rest[CoefficientList[Series[Sum[DivisorSigma[0, k] (Exp[x] - 1)^k/k!, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!]
nmax = 24; Rest[CoefficientList[Series[Sum[DivisorSigma[0, k] x^k/Product[(1 - j x), {j, 1, k}], {k, 1, nmax}], {x, 0, nmax}], x]]
Table[Sum[StirlingS2[n, k] DivisorSigma[0, k], {k, 1, n}], {n, 1, 24}]

G.f.: Sum_{k>=1} tau(k)*x^k / Product_{j=1..k} (1 - j*x).

a(n) = Sum_{k=1..n} Stirling2(n,k)*tau(k).

cp's OEIS Frontend

A330353 Expansion of e.g.f. Sum_{k>=1} (exp(x) - 1)^k / (k * (1 - (exp(x) - 1)^k)).

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