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
A367889
Expansion of e.g.f. exp(3*(exp(x) - 1) + 2*x).
Original entry on oeis.org
1, 5, 28, 173, 1165, 8468, 65923, 546197, 4791214, 44301143, 430158397, 4372004546, 46381674085, 512328076385, 5879362011436, 69958289731457, 861605015493073, 10965899141265500, 144018319806024991, 1949190279770578145, 27153595018237222774
Offset: 0
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nmax = 20; CoefficientList[Series[Exp[3 (Exp[x] - 1) + 2 x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = 2 a[n - 1] + 3 Sum[Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]
Table[Sum[Binomial[n, k] 2^(n - k) BellB[k, 3], {k, 0, n}], {n, 0, 20}]
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my(x='x+O('x^30)); Vec(serlaplace(exp(3*(exp(x) - 1) + 2*x))) \\ Michel Marcus, Dec 04 2023
A366199
Expansion of e.g.f. exp(4*(exp(x) - 1) + 2*x).
Original entry on oeis.org
1, 6, 40, 292, 2308, 19580, 177044, 1696572, 17148916, 182114972, 2024979604, 23506175868, 284125820724, 3567957972316, 46454893734612, 625979771144764, 8715626185644916, 125200337417147932, 1853095248414187796, 28225529312569364732, 441925530173009732532
Offset: 0
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nmax = 20; CoefficientList[Series[Exp[4 (Exp[x] - 1) + 2 x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = 2 a[n - 1] + 4 Sum[Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]
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my(x='x+O('x^30)); Vec(serlaplace(exp(4*(exp(x) - 1) + 2*x))) \\ Michel Marcus, Dec 07 2023
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