A355252
Expansion of e.g.f. exp(2*(exp(x) - 1) + 3*x).
Original entry on oeis.org
1, 5, 27, 157, 979, 6517, 46107, 345261, 2726243, 22623525, 196712171, 1787356765, 16929897395, 166808851541, 1706299041211, 18088031239437, 198392625389315, 2248104026019461, 26283054263021963, 316637825898555069, 3926250785070282579, 50056384077880370101
Offset: 0
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nmax = 25; CoefficientList[Series[Exp[2*Exp[x]-2+3*x], {x, 0, nmax}], x] * Range[0, nmax]!
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my(x='x+O('x^30)); Vec(serlaplace(exp(2*(exp(x) - 1) + 3*x))) \\ Michel Marcus, Dec 04 2023
A367890
Expansion of e.g.f. exp(3*(exp(x) - 1 - x)).
Original entry on oeis.org
1, 0, 3, 3, 30, 93, 633, 3342, 22809, 156063, 1183872, 9453711, 80455125, 721576560, 6809391111, 67332650007, 695777512638, 7493572404345, 83926492573341, 975467527353750, 11744536832206149, 146234590864310019, 1880198749437144456, 24928860500681953683
Offset: 0
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nmax = 23; CoefficientList[Series[Exp[3 (Exp[x] - 1 - x)], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = 3 Sum[Binomial[n - 1, k] a[n - k - 1], {k, 1, n - 1}]; Table[a[n], {n, 0, 23}]
Table[Sum[Binomial[n, k] (-3)^(n - k) BellB[k, 3], {k, 0, n}], {n, 0, 23}]
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my(x='x+O('x^30)); Vec(serlaplace(exp(3*(exp(x) - 1 - x)))) \\ Michel Marcus, Dec 04 2023
A367921
Expansion of e.g.f. exp(4*(exp(x) - 1) - 3*x).
Original entry on oeis.org
1, 1, 5, 17, 93, 505, 3269, 22657, 172461, 1407177, 12284629, 113832273, 1114775869, 11487315481, 124118143717, 1401808691489, 16504815145421, 202101235848297, 2568312461002741, 33808677627863537, 460227870278020957, 6468672644291075001, 93745096205219336709
Offset: 0
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b:= proc(n, k, m) option remember; `if`(n=0, 4^m, `if`(k>0,
b(n-1, k-1, m+1)*k, 0)+m*b(n-1, k, m)+b(n-1, k+1, m))
end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..22); # Alois P. Heinz, Apr 29 2025
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nmax = 22; CoefficientList[Series[Exp[4 (Exp[x] - 1) - 3 x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = -3 a[n - 1] + 4 Sum[Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 22}]
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