A381505 Expansion of e.g.f. exp(2*x/3) / (1-3*x)^(1/9).
1, 1, 2, 10, 88, 1064, 16144, 293968, 6241280, 151328512, 4124855296, 124843943936, 4153947277312, 150699794606080, 5919989155033088, 250339939417452544, 11338037538551824384, 547552961327680913408, 28087260712728645468160, 1525087432592278987866112
Offset: 0
Keywords
Programs
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PARI
my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(2*x/3)/(1-3*x)^(1/9)))
Formula
a(n) = n! * Sum_{k=0..n} (-3)^k * (2/3)^(n-k) * binomial(-1/9,k)/(n-k)!.
a(n) = (3*n-2)*a(n-1) - 2*(n-1)*a(n-2) for n > 1.
a(n) ~ sqrt(2*Pi) * 3^n * n^(n - 7/18) / (Gamma(1/9) * exp(n - 2/9)). - Vaclav Kotesovec, Apr 23 2025