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A386833 a(n) = Sum_{k=0..n} binomial(3*n+1,k) * binomial(3*n-k-1,n-k).

%I A386833 #16 Aug 05 2025 10:04:01
%S A386833 1,6,59,656,7701,93210,1150495,14395428,181936169,2317140014,
%T A386833 29691138099,382334271544,4943464235069,64137141682242,
%U A386833 834561532624967,10886878474010700,142332442919829585,1864423992564121686,24464149489904517211,321499324010641490016,4230840338116037836901
%N A386833 a(n) = Sum_{k=0..n} binomial(3*n+1,k) * binomial(3*n-k-1,n-k).
%F A386833 a(n) = [x^n] (1+x)^(3*n+1)/(1-x)^(2*n).
%F A386833 a(n) = [x^n] 1/((1-x)^2 * (1-2*x)^(2*n)).
%F A386833 a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * (n-k+1) * binomial(3*n+1,k).
%F A386833 a(n) = Sum_{k=0..n} 2^k * (n-k+1) * binomial(2*n+k-1,k).
%o A386833 (PARI) a(n) = sum(k=0, n, binomial(3*n+1, k)*binomial(3*n-k-1, n-k));
%Y A386833 Cf. A116881, A386834.
%Y A386833 Cf. A370097, A386836.
%Y A386833 Cf. A383326.
%K A386833 nonn
%O A386833 0,2
%A A386833 _Seiichi Manyama_, Aug 05 2025