A381484 Expansion of e.g.f. exp(-x/3) / (1-3*x)^(1/9).
1, 0, 1, 6, 57, 708, 10905, 200538, 4287633, 104507496, 2860291089, 86853807630, 2897638853769, 105357244427244, 4146601837761513, 175632278607964962, 7965651564924845985, 385161391574120046672, 19778647046883844762017, 1074979845580061777989014
Offset: 0
Keywords
Programs
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PARI
my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-x/3)/(1-3*x)^(1/9)))
Formula
a(n) = (-1)^n * n! * Sum_{k=0..n} (1/3)^(n-2*k) * binomial(-1/9,k)/(n-k)!.
a(n) = (n-1) * (3*a(n-1) + a(n-2)) for n > 1.
a(n) ~ sqrt(2*Pi) * 3^n * n^(n - 7/18) / (Gamma(1/9) * exp(n + 1/9)). - Vaclav Kotesovec, Apr 23 2025