A383315 Expansion of e.g.f. exp(-x/2) / (1-6*x)^(1/12).
1, 0, 3, 36, 675, 16632, 509085, 18626436, 793001097, 38511087120, 2101009734099, 127215916659540, 8465583820754907, 614101808094096744, 48230098800348987405, 4077120575169267005268, 369111206211249734907345, 35630377583888099367357984, 3653123185073359871950788963
Offset: 0
Keywords
Programs
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PARI
my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-x/2)/(1-6*x)^(1/12)))
Formula
a(n) = (-1)^n * n! * Sum_{k=0..n} 3^k * (1/2)^(n-2*k) * binomial(-1/12,k)/(n-k)!.
a(n) = 3*(n-1) * (2*a(n-1) + a(n-2)) for n > 1.
a(n) ~ Pi * (2 - sqrt(3))^(1/4) * 2^(n + 1/2) * 3^(n - 3/8) * n^(n - 5/12) / (Gamma(1/3) * Gamma(1/4) * exp(n + 1/12)). - Vaclav Kotesovec, Apr 23 2025