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A367980
Expansion of e.g.f. exp(-2*x) / (2 - exp(3*x)).
6
1, 1, 19, 217, 3835, 82801, 2150659, 65156617, 2256029515, 87878584801, 3803459964499, 181078683329017, 9404687464288795, 529155742667806801, 32063235363798322339, 2081586179439325213417, 144148514796485770141675, 10606079719868369436964801, 826272285216863547170504179
OFFSET
0,3
LINKS
FORMULA
a(n) = Sum_{k>=0} (3*k-2)^n / 2^(k+1).
a(n) = (-2)^n + Sum_{k=1..n} binomial(n,k) * 3^k * a(n-k).
a(n) = Sum_{k=0..n} binomial(n,k) * (-2)^(n-k) * 3^k * A000670(k).
MATHEMATICA
nmax = 18; CoefficientList[Series[Exp[-2 x]/(2 - Exp[3 x]), {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = (-2)^n + Sum[Binomial[n, k] 3^k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
PROG
(Magma)
R<x>:=PowerSeriesRing(Rationals(), 40);
Coefficients(R!(Laplace( Exp(-2*x)/(2-Exp(3*x)) ))); // G. C. Greubel, Jun 11 2024
(SageMath)
def A367980_list(prec):
P.<x> = PowerSeriesRing(QQ, prec)
return P( exp(-2*x)/(2-exp(3*x)) ).egf_to_ogf().list()
A367980_list(40) # G. C. Greubel, Jun 11 2024
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Dec 07 2023
STATUS
approved