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
A367919
Expansion of e.g.f. exp(4*(exp(x) - 1) - x).
Original entry on oeis.org
1, 3, 13, 67, 397, 2627, 19085, 150339, 1272205, 11481155, 109852813, 1109011779, 11765211021, 130707706435, 1516160466573, 18314760232771, 229865470694797, 2991427959247939, 40292570823959693, 560791503840522563, 8053114165521427341, 119158887402348541507
Offset: 0
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nmax = 21; CoefficientList[Series[Exp[4 (Exp[x] - 1) - x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = -a[n - 1] + 4 Sum[Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 21}]
A367920
Expansion of e.g.f. exp(4*(exp(x) - 1) - 2*x).
Original entry on oeis.org
1, 2, 8, 36, 196, 1196, 8116, 60108, 481140, 4126540, 37671540, 364068172, 3707910772, 39645022540, 443540780660, 5177560304972, 62903920321140, 793654042136908, 10378403752717940, 140413475790402892, 1962339063781284468, 28287778534523140428, 420059992540347885172
Offset: 0
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nmax = 22; 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, 22}]
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}]
Showing 1-4 of 4 results.