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A367983
Expansion of e.g.f. exp(-3*x) / (2 - exp(4*x)).
6
1, 1, 33, 481, 11457, 329281, 11405793, 460726561, 21270068097, 1104703800961, 63750028379553, 4046761389279841, 280235644230863937, 21023317859012763841, 1698493239420829750113, 147024466409751282556321, 13575133989036437786590977, 1331764937006253524751217921
OFFSET
0,3
LINKS
FORMULA
a(n) = Sum_{k>=0} (4*k-3)^n / 2^(k+1).
a(n) = (-3)^n + Sum_{k=1..n} binomial(n,k) * 4^k * a(n-k).
a(n) = Sum_{k=0..n} binomial(n,k) * (-3)^(n-k) * 4^k * A000670(k).
MATHEMATICA
nmax = 17; CoefficientList[Series[Exp[-3 x]/(2 - Exp[4 x]), {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = (-3)^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(), 30);
Coefficients(R!(Laplace( Exp(-3*x)/(2-Exp(4*x)) ))); // G. C. Greubel, Jun 11 2024
(SageMath)
def A367983_list(prec):
P.<x> = PowerSeriesRing(QQ, prec)
return P( exp(-3*x)/(2-exp(4*x)) ).egf_to_ogf().list()
A367983_list(40) # G. C. Greubel, Jun 11 2024
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Dec 07 2023
STATUS
approved