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

%I #9 Dec 21 2023 10:23:55

%S 1,7,61,679,9589,167167,3488221,84882679,2360477509,73846507567,

%T 2566948755181,98151533290279,4094166835331029,185010377629273567,

%U 9003499122285420541,469450479424436219479,26109404756356597154149,1542883354097286642881167

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

%F a(n) = 4^n + 3 * Sum_{k=1..n} binomial(n,k) * a(n-k).

%F a(n) = (256/81)*A032033(n) - (1/3)*(3^n + (4/3)*2^n + 16/9 + (64/27)*0^n).

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

%o a(n, m=4, 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, A368323.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Dec 21 2023