%I #13 Nov 16 2023 11:51:32

%S 1,1,4,37,505,9076,202519,5397337,167237410,5906437651,234185226409,

%T 10299442379062,497567448860365,26191947311138701,1492143868674196924,

%U 91466801447468838337,6002791454773440431245,419938013721629866235476

%N Expansion of e.g.f. (exp(3*x) / (2 - exp(3*x)))^(1/6).

%F a(n) = Sum_{k=0..n} (-3)^(n-k) * (Product_{j=0..k-1} (6*j+1)) * Stirling2(n,k).

%F a(0) = 1; a(n) = Sum_{k=1..n} (-3)^k * (5/3 * k/n - 2) * binomial(n,k) * a(n-k).

%F a(0) = 1; a(n) = a(n-1) + Sum_{k=1..n-1} 3^k * binomial(n-1,k) * a(n-k).

%o (PARI) a(n) = sum(k=0, n, (-3)^(n-k)*prod(j=0, k-1, 6*j+1)*stirling(n, k, 2));

%Y Cf. A365779.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Nov 16 2023