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A386863 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(3*n+1,k) * binomial(3*n-k-1,n-k).

%I A386863 #9 Aug 06 2025 08:39:41
%S A386863 1,8,117,1948,34283,622272,11519692,216193460,4098365799,78293227384,
%T A386863 1504814127893,29066030323920,563717999500852,10970568626688704,
%U A386863 214125123753359544,4189892211091193380,82166338354628744159,1614453403457943056184,31776198133079795063887
%N A386863 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(3*n+1,k) * binomial(3*n-k-1,n-k).
%F A386863 a(n) = [x^n] (1+x)^(3*n+1)/(1-2*x)^(2*n).
%F A386863 a(n) = [x^n] 1/((1-x)^2 * (1-3*x)^(2*n)).
%F A386863 a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * (n-k+1) * binomial(3*n+1,k).
%F A386863 a(n) = Sum_{k=0..n} 3^k * (n-k+1) * binomial(2*n+k-1,k).
%o A386863 (PARI) a(n) = sum(k=0, n, 2^(n-k)*binomial(3*n+1, k)*binomial(3*n-k-1, n-k));
%Y A386863 Cf. A384950, A386866.
%K A386863 nonn
%O A386863 0,2
%A A386863 _Seiichi Manyama_, Aug 06 2025