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Expansion of e.g.f. exp(3*x) / (4 - 3*exp(x)).
2

%I #10 Dec 21 2023 10:23:59

%S 1,6,48,516,7212,125436,2616348,63662556,1770359772,55384885596,

%T 1925211581148,73613650011996,3070625126631132,138757783222353756,

%U 6752624341715261148,352087859568330751836,19582053567267458627292,1157162515572965014445916

%N Expansion of e.g.f. exp(3*x) / (4 - 3*exp(x)).

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

%F a(n) = (64/27)*A032033(n) - (1/3)*(2^n + 4/3 + (16/9)*0^n).

%o (PARI) b(n, t) = sum(k=0, n, t^k*k!*stirling(n, k, 2));

%o a(n, m=3, t=3) = my(u=1+1/t); u^m*b(n, t)-(1/t)*sum(j=0, m-1, u^j*(m-1-j)^n);

%Y Cf. A032033, A201354, A368322, A368324.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Dec 21 2023