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

%I #10 Dec 21 2023 10:24:16

%S 1,4,22,166,1642,20254,299722,5174446,102094042,2266154014,

%T 55890234922,1516265078926,44874837768442,1438774580904574,

%U 49678366226498122,1837828899444250606,72522300447277154842,3040654011774599283934,134985159308312666889322

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

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

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

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

%o a(n, m=2, 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, A368320, A368321.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Dec 21 2023