A308862
Expansion of e.g.f. 1/(1 - x*(1 + 3*x + x^2)*exp(x)).
Original entry on oeis.org
1, 1, 10, 81, 976, 14505, 258456, 5377897, 127852096, 3419620209, 101625743080, 3322169384721, 118475520287136, 4577175039397753, 190436902905933880, 8489222610046324665, 403657900923994965376, 20393319895130130117729, 1090902632352025316904648
Offset: 0
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nmax = 18; CoefficientList[Series[1/(1 - x (1 + 3 x + x^2) Exp[x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] k^3 a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
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my(x='x+O('x^25)); Vec(serlaplace(1/(1 - x*(1 + 3*x + x^2)*exp(x)))) \\ Michel Marcus, Mar 10 2022
A308946
Expansion of e.g.f. 1/(1 - x*(1 + x/2)*exp(x)).
Original entry on oeis.org
1, 1, 5, 30, 244, 2485, 30351, 432502, 7043660, 129050649, 2627117875, 58829021416, 1437117395946, 38032508860177, 1083932872119839, 33098858988564090, 1078083456543449416, 37309607437056658129, 1367138649165397662627, 52879280631976735387588
Offset: 0
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nmax = 19; CoefficientList[Series[1/(1 - x (1 + x/2) Exp[x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] Binomial[k + 1, 2] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 19}]
A308863
Expansion of e.g.f. (1 + LambertW(-x))/(1 + 2*LambertW(-x)).
Original entry on oeis.org
1, 1, 6, 57, 736, 11985, 235296, 5403937, 142073856, 4206560769, 138483596800, 5017244970441, 198363105460224, 8498001799768273, 392127481640165376, 19388814120804416625, 1022681739669784231936, 57317273018414456262273, 3401527253966521309200384
Offset: 0
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nmax = 18; CoefficientList[Series[(1 + LambertW[-x])/(1 + 2 LambertW[-x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] k^k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
A335577
a(0) = 1; a(n) = -Sum_{k=1..n} binomial(n,k) * k^2 * a(n-k).
Original entry on oeis.org
1, -1, -2, 9, 32, -285, -1236, 18725, 86176, -2087001, -9204580, 351964569, 1336442304, -83422970917, -231889447076, 26389118293005, 35917342192064, -10722110983670193, 5028963509133756, 5432569724760331841, -14852185163192897120, -3352369390318855889661
Offset: 0
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a[0] = 1; a[n_] := a[n] = -Sum[Binomial[n, k] k^2 a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 21}]
nmax = 21; CoefficientList[Series[1/(1 + Exp[x] x (1 + x)), {x, 0, nmax}], x] Range[0, nmax]!
A336183
a(n) = n^2 + (1/n) * Sum_{k=1..n-1} binomial(n,k) * k * a(k) * (n-k)^2.
Original entry on oeis.org
1, 5, 23, 154, 1389, 15636, 211231, 3329264, 59969097, 1215233380, 27362096211, 677690995488, 18310602210445, 535964033279780, 16894811428737495, 570603293774677696, 20556251540382371217, 786832900592755991364, 31889277719673937849243, 1364231113649221829763200
Offset: 1
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a[n_] := a[n] = n^2 + (1/n) Sum[Binomial[n, k] k a[k] (n - k)^2, {k, 1, n - 1}]; Table[a[n], {n, 1, 20}]
nmax = 20; CoefficientList[Series[-Log[1 - Exp[x] x (1 + x)], {x, 0, nmax}], x] Range[0, nmax]! // Rest
A336960
E.g.f.: 1 / (1 - x * (2 + x) * exp(x)).
Original entry on oeis.org
1, 2, 14, 132, 1676, 26590, 506202, 11242952, 285383240, 8149464954, 258575410190, 9024809281972, 343619185754748, 14173557899208422, 629600469603730562, 29965010056866657600, 1521221783964264806672, 82053967063309770102130, 4686301361507067542636694
Offset: 0
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nmax = 18; CoefficientList[Series[1/(1 - x (2 + x) Exp[x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] k (k + 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
Showing 1-6 of 6 results.