A367785
Expansion of e.g.f. exp(exp(3*x) - x - 1).
Original entry on oeis.org
1, 2, 13, 89, 772, 7745, 87949, 1109288, 15332539, 229840361, 3706130914, 63857565095, 1169261937973, 22646779177898, 462143532144937, 9902312863237637, 222119823632283628, 5202170552214520637, 126914730275907871201, 3218552632981994910248, 84686139239808135094879, 2307953474037054591248501
Offset: 0
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nmax = 21; CoefficientList[Series[Exp[Exp[3 x] - x - 1], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = -a[n - 1] + Sum[Binomial[n - 1, k - 1] 3^k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 21}]
Table[Sum[(-1)^(n - k) Binomial[n, k] 3^k BellB[k], {k, 0, n}], {n, 0, 21}]
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my(x='x+O('x^30)); Vec(serlaplace(exp(exp(3*x) - x - 1))) \\ Michel Marcus, Nov 30 2023
A367743
Expansion of e.g.f. exp(1 - x - exp(2*x)).
Original entry on oeis.org
1, -3, 5, 1, -7, -75, -99, 1241, 10161, 18989, -332299, -3857551, -14440151, 141168997, 2807256333, 20182451657, -42073176479, -2999363709091, -38439478980891, -161835672017439, 3439471815545177, 87228227501354517, 937579822282327421, 216540362854403513, -198501712690150659055
Offset: 0
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nmax = 24; CoefficientList[Series[Exp[1 - x - Exp[2 x]], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = -a[n - 1] - Sum[Binomial[n - 1, k - 1] 2^k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 24}]
Table[Sum[(-1)^(n - k) Binomial[n, k] 2^k BellB[k, -1], {k, 0, n}], {n, 0, 24}]
A367784
a(n) = exp(1) * Sum_{k>=0} (-1)^k * (n*k - 1)^n / k!.
Original entry on oeis.org
1, -2, 5, 17, 17, -8151, -311435, -777974, 927723585, 82906687673, 1693962380101, -707005824990631, -137258747025993071, -10253960705018807830, 1697644859939460151413, 803696888217607331079149, 148126297324647875348070657, -323461353221296480463456191
Offset: 0
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Table[n! SeriesCoefficient[Exp[1 - x - Exp[n x]], {x, 0, n}], {n, 0, 17}]
Unprotect[Power]; 0^0 = 1; Table[Sum[(-1)^(n - k) Binomial[n, k] n^k BellB[k, -1], {k, 0, n}], {n, 0, 17}]
Showing 1-3 of 3 results.