A306039 Expansion of e.g.f. Product_{k>=1} 1/(1 - log(1 + x)^k/k!).
1, 1, 2, 3, 14, 0, 359, -1988, 28706, -312210, 4387572, -62769366, 1006242599, -17203315363, 318393704043, -6296931104285, 133039045075494, -2986262905171914, 71018001954178952, -1783064497977512206, 47133484019671647932, -1308274154275749372040, 38042727898691562357962
Offset: 0
Keywords
Links
- N. J. A. Sloane, Transforms
- Eric Weisstein's World of Mathematics, Stirling Transform
Programs
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Maple
a:=series(mul(1/(1-log(1+x)^k/k!),k=1..100),x=0,23): seq(n!*coeff(a,x,n),n=0..22); # Paolo P. Lava, Mar 26 2019
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Mathematica
nmax = 22; CoefficientList[Series[Product[1/(1 - Log[1 + x]^k/k!), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! nmax = 22; CoefficientList[Series[Exp[Sum[Sum[Log[1 + x]^(j k)/(k (j!)^k), {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]! Table[Sum[StirlingS1[n, k] Total[Apply[Multinomial, IntegerPartitions[k], {1}]], {k, 0, n}], {n, 0, 22}]
Formula
E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} log(1 + x)^(j*k)/(k*(j!)^k)).
a(n) = Sum_{k=0..n} Stirling1(n,k)*A005651(k).