A306039 -id:A306039 - cp's OEIS frontend

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

Ilya Gutkovskiy, Jun 17 2018

sign

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

Cf. A007837, A305547, A306039.

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

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}]

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).

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

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Apr 12 2019

nonn

Cf. A005651, A140585, A306039, A320349.

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}]

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

cp's OEIS Frontend

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

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