A305987
Expansion of e.g.f. Product_{k>=1} (1 + (exp(x) - 1)^k/k).
Original entry on oeis.org
1, 1, 2, 9, 52, 355, 2976, 29897, 343988, 4423503, 63088600, 992691205, 17095554444, 319404545291, 6427307831504, 138546745515393, 3185841858310180, 77866726065935239, 2016161715005701128, 55127056896177521981, 1587073087715010466556, 47982707153606476112067, 1519931218769637781731712
Offset: 0
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b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(combinat[multinomial](n, n-i*j, i$j)/j!*
b(n-i*j, i-1)*(i-1)!^j, j=0..min(1, n/i))))
end:
a:= n-> add(Stirling2(n, j)*b(j$2), j=0..n):
seq(a(n), n=0..25); # Alois P. Heinz, Jun 15 2018
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nmax = 22; CoefficientList[Series[Product[(1 + (Exp[x] - 1)^k/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Exp[Sum[Sum[(-1)^(k + 1) (Exp[x] - 1)^(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, (-#)^(1 - k/#) &] b[n - k], {k, 1, n}]; a[n_] := a[n] = Sum[StirlingS2[n, k] b[k], {k, 0, n}]; Table[a[n], {n, 0, 22}]
A347006
E.g.f.: Product_{k>=1} (1 + exp(x) * x^k / k!).
Original entry on oeis.org
1, 1, 3, 10, 43, 206, 1044, 5909, 38371, 272314, 1995208, 14869889, 115433344, 965259881, 8773348601, 84608514095, 837220780691, 8334354200226, 83498917650084, 855936118936073, 9180736840445788, 104439240481045949, 1253608634906635901
Offset: 0
a(3) = 10 counts: (1#,1,1), (1,1#,1), (1,1,1#), (1#,2#,2#), (2#,1#,2#), (2#,2#,1#), (2#,2#,2), (2#,2,2#), (2,2#,2#), (3#,3#,3#). - _John Tyler Rascoe_, Jul 16 2025
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b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
b(n, i-1)+add(b(n-j, min(n-j, i-1))/i!/(j-i)!, j=i..n)))
end:
a:= n-> n!*b(n$2):
seq(a(n), n=0..22); # Alois P. Heinz, Jul 17 2025
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nmax = 22; CoefficientList[Series[Product[(1 + Exp[x] x^k/k!), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
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C_x(N) = {my(x='x+O('x^(N+1))); Vec(serlaplace(prod(k=1,N, 1 + exp(x)*x^k/k!)))} \\ John Tyler Rascoe, Jul 16 2025
A306040
Expansion of e.g.f. Product_{k>=1} (1 + log(1 + x)^k/k!).
Original entry on oeis.org
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
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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
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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}]
A345759
E.g.f.: Product_{k>=1} (1 - (exp(x) - 1)^k / k!).
Original entry on oeis.org
1, -1, -2, -2, 7, 78, 513, 2665, 9406, -13902, -789143, -11806456, -140040408, -1463842226, -13377115923, -95264642343, -198034245627, 11021440199748, 322964047973519, 6617250866231379, 118668721540190350, 1965786734149801960, 30348547043773563767
Offset: 0
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