A308554 -id:A308554 - cp's OEIS frontend

A330351 Expansion of e.g.f. -Sum_{k>=1} log(1 - (exp(x) - 1)^k) / k.

1, 3, 11, 57, 359, 2793, 25871, 273297, 3268199, 44132313, 659178431, 10710083937, 189256343639, 3636935896233, 75228664345391, 1657133255788977, 38770903634692679, 964609458391250553, 25470259163197390751, 709595190213796188417

Offset: 1

Ilya Gutkovskiy, Dec 11 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..420
Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)

Cf. A000005, A000629, A002746, A008277, A028342, A308554, A318249, A330352, A330353, A330354, A330445.

nmax = 20; CoefficientList[Series[-Sum[Log[1 - (Exp[x] - 1)^k]/k, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
Table[Sum[StirlingS2[n, k] (k - 1)! DivisorSigma[0, k], {k, 1, n}], {n, 1, 20}]

E.g.f.: Sum_{i>=1} Sum_{j>=1} (exp(x) - 1)^(i*j) / (i*j).
E.g.f.: log(Product_{k>=1} 1 / (1 - (exp(x) - 1)^k)^(1/k)).
G.f.: Sum_{k>=1} (k - 1)! * tau(k) * x^k / Product_{j=1..k} (1 - j*x), where tau = A000005.
a(n) = Sum_{k=1..n} Stirling2(n,k) * (k - 1)! * tau(k).
a(n) ~ n! * (log(n) + 2*gamma - log(2) - log(log(2))) / (n * (log(2))^n), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Dec 14 2019

A308555 Expansion of e.g.f. Sum_{k>=1} sigma(k)*(exp(x) - 1)^k/k!, where sigma = sum of divisors (A000203).

Original entry on oeis.org

1, 4, 14, 53, 222, 1011, 4944, 25884, 144963, 865556, 5477661, 36518635, 255323564, 1867122987, 14259709474, 113593734317, 942317654779, 8123227487723, 72599829900774, 671199117610868, 6407156027307909, 63061416571124056, 639303956718643041, 6670690645674913424

Offset: 1

Ilya Gutkovskiy, Jun 07 2019

nonn

Stirling transform of A000203.

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

Cf. A000203, A185003, A308554.

b:= proc(n, m) option remember; uses numtheory;
     `if`(n=0, sigma(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 03 2021

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

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

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

cp's OEIS Frontend

A330351 Expansion of e.g.f. -Sum_{k>=1} log(1 - (exp(x) - 1)^k) / k.

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