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A330449
Expansion of e.g.f. Sum_{k>=1} (exp(x) - 1)^k / (k * (1 - (exp(x) - 1)^k)^2).
3
1, 6, 36, 282, 2460, 25506, 299796, 3921882, 56977740, 913248786, 15917884356, 299358495882, 6066180049020, 131932872768066, 3057940695635316, 75151035318996282, 1954299203147952300, 53684552455571903346, 1553161560008013680676, 47162101103528811791082
OFFSET
1,2
LINKS
FORMULA
E.g.f.: -Sum_{k>=1} k * log(1 - (exp(x) - 1)^k).
E.g.f.: A(x) = log(B(x)), where B(x) = e.g.f. of A306046.
G.f.: Sum_{k>=1} (k - 1)! * sigma_2(k) * x^k / Product_{j=1..k} (1 - j*x), where sigma_2 = A001157.
exp(Sum_{n>=1} a(n) * log(1 + x)^n / n!) = g.f. of A000219.
a(n) = Sum_{k=1..n} Stirling2(n,k) * (k - 1)! * sigma_2(k).
a(n) ~ n! * zeta(3) * n / (4 * (log(2))^(n+2)). - Vaclav Kotesovec, Dec 15 2019
MATHEMATICA
nmax = 20; CoefficientList[Series[Sum[(Exp[x] - 1)^k/(k (1 - (Exp[x] - 1)^k)^2), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
Table[Sum[StirlingS2[n, k] (k - 1)! DivisorSigma[2, k], {k, 1, n}], {n, 1, 20}]
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Dec 15 2019
STATUS
approved