A336951
E.g.f.: 1 / (1 - x * exp(3*x)).
Original entry on oeis.org
1, 1, 8, 69, 780, 11145, 191178, 3823785, 87406056, 2247785073, 64228084110, 2018771719569, 69221032558956, 2571290056399545, 102860527370221026, 4408690840306136505, 201557641172689004112, 9790792086366911655009, 503570143277542340304534
Offset: 0
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nmax = 18; CoefficientList[Series[1/(1 - x Exp[3 x]), {x, 0, nmax}], x] Range[0, nmax]!
Join[{1}, Table[n! Sum[(3 (n - k))^k/k!, {k, 0, n}], {n, 1, 18}]]
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] k 3^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
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seq(n)={ Vec(serlaplace(1 / (1 - x*exp(3*x + O(x^n))))) } \\ Andrew Howroyd, Aug 08 2020
A336947
E.g.f.: 1 / (exp(-2*x) - x).
Original entry on oeis.org
1, 3, 14, 98, 920, 10792, 151888, 2494032, 46803072, 988095104, 23178247424, 598074306304, 16835199087616, 513385352524800, 16859837094942720, 593234633904293888, 22265289445252628480, 887889931920920313856, 37489832605652634763264, 1670894259596134872711168
Offset: 0
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nmax = 19; CoefficientList[Series[1/(Exp[-2 x] - x), {x, 0, nmax}], x] Range[0, nmax]!
Table[n! Sum[(2 (n - k + 1))^k/k!, {k, 0, n}], {n, 0, 19}]
a[0] = 1; a[n_] := a[n] = 3 n a[n - 1] - Sum[Binomial[n, k] (-2)^k a[n - k], {k, 2, n}]; Table[a[n], {n, 0, 19}]
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seq(n)={ Vec(serlaplace(1 / (exp(-2*x + O(x*x^n)) - x))) } \\ Andrew Howroyd, Aug 08 2020
A336949
a(n) = n! * [x^n] 1 / (exp(-n*x) - x).
Original entry on oeis.org
1, 2, 14, 195, 4440, 147745, 6698448, 394852577, 29250137472, 2652483234033, 288363456748800, 36952298766628465, 5504130616452258816, 941845623036360908489, 183298110723156455921664, 40221612394630225987208625, 9876429434585097671993032704
Offset: 0
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Table[n! SeriesCoefficient[1/(Exp[-n x] - x), {x, 0, n}], {n, 0, 16}]
Join[{1}, Table[n! Sum[(n (n - k + 1))^k/k!, {k, 0, n}], {n, 1, 16}]]
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a(n)={n!*polcoef(1/(exp(-n*x + O(x*x^n)) - x), n)} \\ Andrew Howroyd, Aug 08 2020
A368292
Expansion of e.g.f. exp(x) / (1 - x*exp(3*x)).
Original entry on oeis.org
1, 2, 11, 97, 1109, 15821, 271255, 5425967, 124032137, 3189663433, 91141189931, 2864686414547, 98226339773677, 3648723468367277, 145961603607355775, 6256040115149251591, 286015221831125756945, 13893373399551594686609, 714578347872583574806099
Offset: 0
A368237
Expansion of e.g.f. 1/(exp(-x) - 3*x).
Original entry on oeis.org
1, 4, 31, 361, 5605, 108781, 2533447, 68836279, 2137543177, 74673228457, 2898494302651, 123757822391083, 5764497138070381, 290878956151681405, 15806942065094830735, 920336494043393536591, 57157621592164505969425, 3771643127452655490322513
Offset: 0
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a[n_] := n! Sum[3^(n - k) (n - k + 1)^k / k!, {k, 0, n}];Table[a[n],{n,0,17}] (* or *) a[0] = 1;a[n_] := 3n a[n - 1] + Sum[(-1)^(k - 1) Binomial[n, k] a[n - k], {k, 1, n}];Table[a[n],{n,0,17}] (* James C. McMahon, Dec 18 2023 *)
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a(n) = n!*sum(k=0, n, 3^(n-k)*(n-k+1)^k/k!);
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my(x='x+O('x^25)); Vec(serlaplace(1/(exp(-x) - 3*x))) \\ Michel Marcus, Dec 18 2023
A368293
Expansion of e.g.f. exp(2*x) / (1 - x*exp(3*x)).
Original entry on oeis.org
1, 3, 16, 137, 1572, 22457, 384934, 7699449, 176004376, 4526214641, 129331581954, 4065059876033, 139385578253524, 5177627842411065, 207123084913936174, 8877473910719477033, 405862594146337680816, 19715036609777115714401, 1014004150296746677804666
Offset: 0
Showing 1-6 of 6 results.