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

%I #9 Dec 21 2023 10:24:12

%S 1,5,31,245,2455,30365,449551,7761605,153140935,3399230765,

%T 83835351871,2274397617365,67312256650615,2158161871352765,

%U 74517549339738991,2756743349166359525,108783450670915699495,4560981017661898860365,202477738962469000202911

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

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

%F a(n) = (27/8)*A004123(n+1) - (1/2)*(2^n + 3/2 + (9/4)*0^n).

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

%o a(n, m=3, t=2) = 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. A004123, A201339, A368319, A368321.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Dec 21 2023