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Expansion of the e.g.f. (exp(x) / (3 - 2*exp(x)))^(2/3).
0

%I #11 Nov 15 2023 08:03:44

%S 1,2,8,52,468,5372,74948,1230812,23251908,496661532,11834467588,

%T 311195370972,8950935130948,279540192840092,9419760953149828,

%U 340658973061341532,13160048773006619588,540850933969855649052,23561995002376443953668

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

%F a(n) = Sum_{k=0..n} (-1)^(n-k) * (Product_{j=0..k-1} (3*j+2)) * Stirling2(n,k).

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

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

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

%Y Cf. A136727, A201339.

%Y Cf. A365558.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Nov 15 2023