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A370716
a(n) = 3^(2*n) * [x^n] Product_{k>=1} (1 + 2*x^k)^(1/3).
6
1, 6, 18, 1170, -1890, 133326, 101250, 20498994, -164656314, 3778220862, -28085954094, 771567716970, -10691904063114, 183594050113518, -2711145260068326, 49416883617381354, -789899109743435994, 13176840267952166070, -216403389726994588086, 3681309971143060236810
OFFSET
0,2
FORMULA
G.f.: Product_{k>=1} (1 + 2*(9*x)^k)^(1/3).
a(n) ~ (-1)^(n+1) * c * 18^n / n^(4/3), where c = QPochhammer(-1/2)^(1/3) / (3*Gamma(2/3)) = 0.2623638446186535909018671540030519...
MATHEMATICA
nmax = 20; CoefficientList[Series[Product[(1 + 2*x^k), {k, 1, nmax}]^(1/3), {x, 0, nmax}], x] * 3^(2*Range[0, nmax])
nmax = 20; CoefficientList[Series[Product[(1 + 2*(9*x)^k), {k, 1, nmax}]^(1/3), {x, 0, nmax}], x]
nmax = 20; CoefficientList[Series[(QPochhammer[-2, x]/3)^(1/3), {x, 0, nmax}], x] * 3^(2*Range[0, nmax])
CROSSREFS
KEYWORD
sign
AUTHOR
Vaclav Kotesovec, Feb 27 2024
STATUS
approved