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Expansion of e.g.f. Sum_{k>=0} exp((3^k - 1)*x) * x^k/k!.
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%I #17 Feb 26 2023 06:56:17

%S 1,1,5,37,521,12361,510605,35837677,4348414481,903630399121,

%T 325415100648725,201805338104622517,217331913727442676761,

%U 404193405278758441895641,1306527408146744068362681245,7302236837745565755664036677757

%N Expansion of e.g.f. Sum_{k>=0} exp((3^k - 1)*x) * x^k/k!.

%F G.f.: Sum_{k>=0} x^k/(1 - (3^k - 1)*x)^(k+1).

%F a(n) = Sum_{k=0..n} (3^k - 1)^(n-k) * binomial(n,k).

%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(1+sum(k=1, N, exp((3^k-1)*x)*x^k/k!)))

%o (PARI) my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, x^k/(1-(3^k-1)*x)^(k+1)))

%o (PARI) a(n) = sum(k=0, n, (3^k-1)^(n-k)*binomial(n, k));

%Y Cf. A001831, A360934, A360935.

%Y Cf. A135079, A135753.

%K nonn,easy

%O 0,3

%A _Seiichi Manyama_, Feb 26 2023