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A357543 a(n) = (3*n+1)!/(3^n*n!) * Product_{k=1..n} (3*k - 2), for n >= 0.

%I A357543 #8 Oct 11 2022 00:50:00
%S A357543 1,8,1120,627200,896896000,2611761152000,13497581633536000,
%T A357543 112839782456360960000,1427423248072966144000000,
%U A357543 25979103114927983820800000000,653945983608967208737177600000000,22056290135163246016287526092800000000,971138454651237722097139773865984000000000
%N A357543 a(n) = (3*n+1)!/(3^n*n!) * Product_{k=1..n} (3*k - 2), for n >= 0.
%C A357543 Equals row sums of triangle A357540.
%C A357543 a(n) = (3*n+1) * A178575(n) for n >= 0.
%F A357543 E.g.f.: Sum_{n>=0} a(n) * x^(3*n+1) / (3*n+1)! = x/(1 - x^3)^(1/3).
%F A357543 a(n) ~ sqrt(2*Pi) * 3^(3*n + 3/2) * n^(3*n + 5/6) / (Gamma(1/3) * exp(3*n)). - _Vaclav Kotesovec_, Oct 10 2022
%o A357543 (PARI) {a(n) = (3*n+1)!/(3^n*n!) * prod(k=1, n, 3*k-2)}
%o A357543 for(n=0,20, print1(a(n),", "))
%Y A357543 Cf. A357540, A178575, A004117.
%K A357543 nonn
%O A357543 0,2
%A A357543 _Paul D. Hanna_, Oct 10 2022