A307525 Expansion of e.g.f. Product_{k>=1} 1/(1 - log(1/(1 - x))^k/k!).

1, 1, 4, 21, 146, 1240, 12479, 144970, 1908682, 28079550, 456458832, 8125189974, 157190542607, 3284222304545, 73705849847317, 1768479436456975, 45180024672023814, 1224529894981726614, 35096983241255523572, 1060703070504583747430, 33714045363258013414692

Offset

0,3

Links

Table of n, a(n) for n=0..20.

Formula

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

a(n) = Sum_{k=0..n} |Stirling1(n,k)|*A005651(k).

a(n) ~ c * sqrt(2*Pi) * n^(n + 1/2) / (exp(1) - 1)^(n+1), where c = A247551 = Product_{k>=2} 1/(1-1/k!). - Vaclav Kotesovec, Apr 13 2019

Mathematica

nmax = 20; CoefficientList[Series[Product[1/(1 - Log[1/(1 - x)]^k/k!), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 20; CoefficientList[Series[Exp[Sum[Sum[Log[1/(1 - x)]^(j k)/(k (j!)^k), {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Abs[StirlingS1[n, k]] Total[Apply[Multinomial, IntegerPartitions[k], {1}]], {k, 0, n}], {n, 0, 20}]

Crossrefs

Cf. A005651, A140585, A306039, A320349.

Sequence in context: A305986 A233481 A308337 * A345749 A327872 A163861

Adjacent sequences: A307522 A307523 A307524 * A307526 A307527 A307528

Keyword

nonn

Author

Ilya Gutkovskiy, Apr 12 2019

Status

approved