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Expansion of e.g.f. exp(x * (2 + exp(x))).
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%I #15 Feb 03 2024 10:15:20

%S 1,3,11,48,241,1358,8445,57256,419233,3290202,27507349,243731084,

%T 2278919697,22402234390,230781192301,2484462888312,27880896280513,

%U 325432611292082,3943062342781605,49504837209940612,642982531293731761,8626753575445207278

%N Expansion of e.g.f. exp(x * (2 + exp(x))).

%F G.f.: Sum_{k>=0} x^k / (1 - (k+2)*x)^(k+1).

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

%F a(0) = 1; a(n) = 2 * a(n-1) + Sum_{k=1..n} binomial(n-1,k-1) * k * a(n-k). - _Ilya Gutkovskiy_, Feb 02 2024

%o (PARI) a(n) = sum(k=0, n, (k+2)^(n-k)*binomial(n, k));

%Y Cf. A000248, A080108, A367875.

%Y Cf. A240165.

%K nonn,easy

%O 0,2

%A _Seiichi Manyama_, Dec 03 2023