A320083 Expansion of e.g.f. Sum_{k>=0} log(1 + k*x)^k.

1, 1, 7, 116, 3574, 177094, 12873962, 1290494904, 170592253320, 28753159552272, 6018433850602848, 1531605185388897552, 465706857941949607008, 166746568516127626614288, 69440517484503283491716400, 33278913978673363553703249408, 18185279212166784139689388753536, 11239676837467731657648932618952576

Offset: 0

Author

Ilya Gutkovskiy, Oct 05 2018

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..235

Crossrefs

Cf. A048594, A048994, A122399, A317172, A320096.

Programs

Maple

1,seq(n!*coeff(series(add( log(1 + k*x)^k,k=1..100), x=0, 18), x, n), n=1..17); # Paolo P. Lava, Jan 09 2019

Mathematica

nmax = 17; CoefficientList[Series[1 + Sum[Log[1 + k x]^k, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
Join[{1}, Table[Sum[StirlingS1[n, k] k! k^n, {k, n}], {n, 17}]]

Formula

a(n) = Sum_{k=0..n} Stirling1(n,k)*k!*k^n.

a(n) ~ c * d^n * n^(2*n + 1/2), where d = 0.298212940253960977992575968955431001807757948758929... and c = 3.40415549717199390989204785905061856492539214306... - Vaclav Kotesovec, Oct 05 2018