A344037
Expansion of e.g.f.: exp(-2*x) / (2 - exp(x)).
Original entry on oeis.org
1, -1, 3, -1, 27, 119, 1203, 11759, 136587, 1771559, 25562403, 405657119, 7022893947, 131714582999, 2660335750803, 57570797728079, 1328913670528107, 32592691757218439, 846383665814342403, 23200396829831840639, 669421949061096575067, 20281206249626017421879
Offset: 0
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R:=PowerSeriesRing(Rationals(), 40);
Coefficients(R!(Laplace( Exp(-2*x)/(2-Exp(x)) ))); // G. C. Greubel, Jun 11 2024
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nmax = 21; CoefficientList[Series[Exp[-2 x]/(2 - Exp[x]), {x, 0, nmax}], x] Range[0, nmax]!
Table[HurwitzLerchPhi[1/2, -n, -2]/2, {n, 0, 21}]
a[n_] := a[n] = (-2)^n + Sum[Binomial[n, k] a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 21}]
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def A344037_list(prec):
P. = PowerSeriesRing(QQ, prec)
return P( exp(-2*x)/(2-exp(x)) ).egf_to_ogf().list()
A344037_list(40) # G. C. Greubel, Jun 11 2024
A292916
a(n) = n! * [x^n] exp(n*x)/(2 - exp(x)).
Original entry on oeis.org
1, 2, 11, 94, 1083, 15666, 272451, 5532206, 128409707, 3352959850, 97259891163, 3102552150006, 107936130271899, 4066743353318114, 164961642651034547, 7167348523420169278, 332081754670735087275, 16343667009638859878298, 851478575825591156040843, 46814697307371602567813126
Offset: 0
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R:=PowerSeriesRing(Rationals(), 50);
A292916:= func< n | Coefficient(R!(Laplace( Exp(n*x)/(2-Exp(x)) )), n) >;
[A292916(n): n in [0..30]]; // G. C. Greubel, Jun 12 2024
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b:= proc(n, k) option remember; k^n +add(
binomial(n, j)*b(j, k), j=0..n-1)
end:
a:= n-> b(n$2):
seq(a(n), n=0..20); # Alois P. Heinz, Sep 27 2017
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Table[n! SeriesCoefficient[Exp[n x]/(2 - Exp[x]), {x, 0, n}], {n, 0, 19}]
Table[HurwitzLerchPhi[1/2, -n, n]/2, {n, 0, 19}]
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a000670(n) = sum(k=0, n, k!*stirling(n, k, 2));
a(n) = 2^n*a000670(n)-sum(k=0, n-1, 2^k*(n-1-k)^n); \\ Seiichi Manyama, Dec 25 2023
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[factorial(n)*( exp(n*x)/(2-exp(x)) ).series(x,n+1).list()[n] for n in (0..30)] # G. C. Greubel, Jun 12 2024
A346208
Expansion of e.g.f.: exp(-3*x) / (2 - exp(x)).
Original entry on oeis.org
1, -2, 6, -14, 54, -62, 966, 4786, 71574, 875938, 12810726, 202739986, 3511712694, 65856494338, 1330170266886, 28785391689586, 664456856787414, 16296345814039138, 423191833100881446, 11600198414334789586, 334710974532291679734, 10140603124807778534338
Offset: 0
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R:=PowerSeriesRing(Rationals(), 40);
Coefficients(R!(Laplace( Exp(-3*x)/(2-Exp(x)) ))); // G. C. Greubel, Jun 11 2024
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nmax = 21; CoefficientList[Series[Exp[-3 x]/(2 - Exp[x]), {x, 0, nmax}], x] Range[0, nmax]!
Table[HurwitzLerchPhi[1/2, -n, -3]/2, {n, 0, 21}]
a[n_] := a[n] = (-3)^n + Sum[Binomial[n, k] a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 21}]
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def A346208_list(prec):
P. = PowerSeriesRing(QQ, prec)
return P( exp(-3*x)/(2-exp(x)) ).egf_to_ogf().list()
A346208_list(40) # G. C. Greubel, Jun 11 2024
Showing 1-3 of 3 results.
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