A308861
Expansion of e.g.f. 1/(1 - x*(1 + x)*exp(x)).
Original entry on oeis.org
1, 1, 6, 39, 352, 3965, 53556, 844123, 15204960, 308118105, 6937562980, 171826160231, 4642588564032, 135891789038629, 4283619809941668, 144674451274329075, 5211965027738046016, 199498704931954788785, 8085413817213212761668, 345895984008645703002559
Offset: 0
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nmax = 19; CoefficientList[Series[1/(1 - x (1 + x) Exp[x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] k^2 a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 19}]
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my(x='x+O('x^25)); Vec(serlaplace(1/(1 - x*(1 + x)*exp(x)))) \\ Michel Marcus, Mar 10 2022
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}]
A335578
a(0) = 1; a(n) = -Sum_{k=1..n} binomial(n,k) * k^3 * a(n-k).
Original entry on oeis.org
1, -1, -6, 15, 272, -745, -29976, 61271, 6065856, -2723697, -1941455080, -3989345041, 897021218400, 4964061925511, -562221881675832, -5689641396555705, 456732442022509184, 7321841133968133023, -464200472167634521800, -10961686347887871324289, 573373115861405030522400
Offset: 0
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a[0] = 1; a[n_] := a[n] = -Sum[Binomial[n, k] k^3 a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]
nmax = 20; CoefficientList[Series[1/(1 + Exp[x] x (1 + 3 x + x^2)), {x, 0, nmax}], x] Range[0, nmax]!
A336184
a(n) = n^3 + (1/n) * Sum_{k=1..n-1} binomial(n,k) * k * a(k) * (n-k)^3.
Original entry on oeis.org
1, 9, 53, 466, 5569, 82656, 1474045, 30664656, 729036801, 19499288680, 579487528861, 18943592776032, 675568129695601, 26099852672860344, 1085904530481561645, 48407032164910589056, 2301727955153266523521, 116286277045753464506568, 6220517619913795356269725
Offset: 1
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a[n_] := a[n] = n^3 + (1/n) Sum[Binomial[n, k] k a[k] (n - k)^3, {k, 1, n - 1}]; Table[a[n], {n, 1, 19}]
nmax = 19; CoefficientList[Series[-Log[1 - Exp[x] x (1 + 3 x + x^2)], {x, 0, nmax}], x] Range[0, nmax]! // Rest
A352357
Expansion of e.g.f. 1/(1 - Sum_{k>=1} binomial(k+2,3) * x^k/k!).
Original entry on oeis.org
1, 1, 6, 40, 364, 4155, 56836, 907158, 16547896, 339587445, 7743161740, 194212763756, 5314051343932, 157520046898695, 5028409083962824, 171984217743856890, 6274444932921616176, 243215342466576246185, 9982290554423689511124, 432464578359391409082952
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-(x+x^2+x^3/6)*exp(x))))
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a(n) = if(n==0, 1, sum(k=1, n, binomial(k+2, 3)*binomial(n, k)*a(n-k)));
Showing 1-5 of 5 results.