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%I #12 Jan 09 2019 09:13:53
%S 1,1,7,116,3574,177094,12873962,1290494904,170592253320,
%T 28753159552272,6018433850602848,1531605185388897552,
%U 465706857941949607008,166746568516127626614288,69440517484503283491716400,33278913978673363553703249408,18185279212166784139689388753536,11239676837467731657648932618952576
%N Expansion of e.g.f. Sum_{k>=0} log(1 + k*x)^k.
%H Vaclav Kotesovec, <a href="/A320083/b320083.txt">Table of n, a(n) for n = 0..235</a>
%F a(n) = Sum_{k=0..n} Stirling1(n,k)*k!*k^n.
%F a(n) ~ c * d^n * n^(2*n + 1/2), where d = 0.298212940253960977992575968955431001807757948758929... and c = 3.40415549717199390989204785905061856492539214306... - _Vaclav Kotesovec_, Oct 05 2018
%p 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
%t nmax = 17; CoefficientList[Series[1 + Sum[Log[1 + k x]^k, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
%t Join[{1}, Table[Sum[StirlingS1[n, k] k! k^n, {k, n}], {n, 17}]]
%Y Cf. A048594, A048994, A122399, A317172, A320096.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Oct 05 2018