A365794 Expansion of e.g.f. 1 / (3 - 2 * exp(2*x))^(3/4).
1, 3, 27, 369, 6849, 160803, 4566987, 152204769, 5822610849, 251445000483, 12098060349147, 641736701136369, 37204969609266849, 2340437711290748163, 158770522442243864907, 11553653430580844747169, 897732793887437892390849, 74182365989862425679675843
Offset: 0
Keywords
Programs
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Mathematica
With[{nn=20},CoefficientList[Series[1/(3-2Exp[2x])^(3/4),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, May 30 2024 *) -
PARI
a(n) = sum(k=0, n, 2^(n-k)*prod(j=0, k-1, 4*j+3)*stirling(n, k, 2));
Formula
a(n) = Sum_{k=0..n} 2^(n-k) * (Product_{j=0..k-1} (4*j+3)) * Stirling2(n,k).
a(0) = 1; a(n) = Sum_{k=1..n} 2^k * (2 - 1/2 * k/n) * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = 3*a(n-1) - 3*Sum_{k=1..n-1} (-2)^k * binomial(n-1,k) * a(n-k).