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A367982
Expansion of e.g.f. exp(-2*x) / (2 - exp(4*x)).
6
1, 2, 36, 584, 13584, 391712, 13563456, 547900544, 25294512384, 1313721631232, 75811987301376, 4812437436975104, 333258221996150784, 25001079178900938752, 2019860245103282896896, 174842541533954981003264, 16143645926877401603702784, 1583744338598987290588086272
OFFSET
0,2
LINKS
FORMULA
a(n) = Sum_{k>=0} (4*k-2)^n / 2^(k+1).
a(n) = (-2)^n + Sum_{k=1..n} binomial(n,k) * 4^k * a(n-k).
a(n) = Sum_{k=0..n} binomial(n,k) * (-2)^(n+k) * A000670(k).
MATHEMATICA
nmax = 17; CoefficientList[Series[Exp[-2 x]/(2 - Exp[4 x]), {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = (-2)^n + Sum[Binomial[n, k] 4^k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 17}]
PROG
(Magma)
R<x>:=PowerSeriesRing(Rationals(), 40);
Coefficients(R!(Laplace( Exp(-2*x)/(2-Exp(4*x)) ))); // G. C. Greubel, Jun 11 2024
(SageMath)
def A367982_list(prec):
P.<x> = PowerSeriesRing(QQ, prec)
return P( exp(-2*x)/(2-exp(4*x)) ).egf_to_ogf().list()
A367982_list(40) # G. C. Greubel, Jun 11 2024
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Dec 07 2023
STATUS
approved