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A327412 a(n) = multinomial(3*n+2; 2, 3, 3, ..., 3) (n times '3').

%I A327412 #20 Apr 19 2025 06:10:44
%S A327412 1,10,560,92400,33633600,22870848000,26072766720000,46174869861120000,
%T A327412 120054661638912000000,438679733628584448000000,
%U A327412 2175851478797778862080000000,14240947928731462652313600000000,120136636726778618934917529600000000,1280656547507460077846220865536000000000
%N A327412 a(n) = multinomial(3*n+2; 2, 3, 3, ..., 3) (n times '3').
%F A327412 a(n) = 2^(-n-1)*3^(-n)*Gamma(3*n + 3).
%F A327412 a(n) = (9*(n-1)^3 + 36*(n-1)^2 + 47*n - 27)*a(n-1)/2 for n > 0.
%F A327412 a(n) / n! = A025035(n+1).
%F A327412 a(n)*(n+1) = A014606(n+1).
%p A327412 a:= n-> combinat[multinomial](3*n+2, 3$n, 2):
%p A327412 seq(a(n), n=0..17);  # _Alois P. Heinz_, Sep 07 2019
%t A327412 multinomial[n_, k_List] := n!/Times @@ (k!);
%t A327412 a[n_] := multinomial[3n+2, Join[{2}, Table[3, {n}]]];
%t A327412 Table[a[n], {n, 0, 17}] (* _Jean-François Alcover_, Apr 19 2025 *)
%o A327412 (SageMath)
%o A327412 def a(n): return multinomial([2] + [3] * n)
%o A327412 [a(n) for n in range(15)]
%Y A327412 Cf. A014606, A327411, A025035.
%K A327412 nonn
%O A327412 0,2
%A A327412 _Peter Luschny_, Sep 07 2019