A383317 Expansion of e.g.f. exp(x/2) / (1-6*x)^(1/12).
1, 1, 4, 46, 838, 20398, 619768, 22564252, 957247708, 46363595644, 2524152072304, 152582368541224, 10139721673875976, 734706716925462184, 57646381491830349472, 4869084744694710293392, 440492624600086270972432, 42494068518463022190243088, 4354423933547086885775444032
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) = n! * Sum_{k=0..n} (-3)^k * (1/2)^(n-2*k) * binomial(-1/12,k)/(n-k)!.
a(n) = (6*n-5)*a(n-1) - 3*(n-1)*a(n-2) for n > 1.
From Vaclav Kotesovec, Apr 23 2025: (Start)
a(n) ~ (sqrt(3) - 1) * 2^(n-1) * 3^n * n^(n - 5/12) * Gamma(11/12) / (sqrt(Pi) * exp(n - 1/12)).
Equivalently, 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)). (End)