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a(0) = 1; a(n) = 3 * Sum_{k=1..n} binomial(n,k) * a(k-1).
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%I #18 Jun 07 2021 14:55:19

%S 1,3,15,81,489,3237,23211,178707,1467051,12768345,117263829,

%T 1131901521,11444383251,120847326879,1329303053391,15197269729689,

%U 180211641841353,2212525627591533,28078380387448515,367782119667874083,4965441830591976339,69014083524412401873,986364827548578356421

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

%F G.f. A(x) satisfies: A(x) = 1 + 3 * x * A(x/(1 - x)) / (1 - x)^2.

%t a[0] = 1; a[n_] := a[n] = 3 Sum[Binomial[n, k] a[k - 1], {k, 1, n}]; Table[a[n], {n, 0, 22}]

%t nmax = 22; A[_] = 0; Do[A[x_] = 1 + 3 x A[x/(1 - x)]/(1 - x)^2 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

%Y Cf. A027710, A032033, A040027, A078940, A343523, A344735, A344840, A345077, A345078, A345081.

%K nonn

%O 0,2

%A _Ilya Gutkovskiy_, Jun 07 2021