A306040 Expansion of e.g.f. Product_{k>=1} (1 + log(1 + x)^k/k!).

1, 1, 0, 3, -14, 80, -479, 3024, -19802, 129114, -740172, 1569150, 66402733, -2071117841, 47637088903, -1025499372275, 21965630892954, -479007175968646, 10740252324008140, -248679637522491506, 5956075167341406752, -147619197124939380988, 3784960110596636657926

Offset: 0

Author

Ilya Gutkovskiy, Jun 17 2018

Keywords

• sign

Links

• N. J. A. Sloane, Transforms
• Eric Weisstein's World of Mathematics, Stirling Transform

Crossrefs

Cf. A007837, A305547, A306039.

Programs

Maple

a:=series(mul(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

Mathematica

nmax = 22; CoefficientList[Series[Product[(1 + Log[1 + x]^k/k!), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Exp[Sum[Sum[(-1)^(k + 1) Log[1 + x]^(j k)/(k (j!)^k), {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
b[0] = 1; b[n_] := b[n] = Sum[(n - 1)!/(n - k)! DivisorSum[k, -# (-#!)^(-k/#) &] b[n - k], {k, 1, n}]; a[n_] := a[n] = Sum[StirlingS1[n, k] b[k], {k, 0, n}]; Table[a[n], {n, 0, 22}]

Formula

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

a(n) = Sum_{k=0..n} Stirling1(n,k)*A007837(k).