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A386896 a(n) = Sum_{k=0..n} binomial(5*n+1,k) * binomial(3*n-k,n-k).

%I A386896 #17 Aug 21 2025 10:06:06
%S A386896 1,9,125,1932,31365,523809,8910356,153544680,2671398309,46822319115,
%T A386896 825501663525,14623742203200,260088366645900,4641248247561324,
%U A386896 83059406374007720,1490097583932329232,26790218420643034533,482571492068274975135,8707190579448431827991
%N A386896 a(n) = Sum_{k=0..n} binomial(5*n+1,k) * binomial(3*n-k,n-k).
%F A386896 a(n) = [x^n] (1+x)^(5*n+1)/(1-x)^(2*n+1).
%F A386896 a(n) = [x^n] 1/((1-x) * (1-2*x))^(2*n+1).
%F A386896 a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(5*n+1,k) * binomial(3*n-k,n-k).
%F A386896 a(n) = Sum_{k=0..n} 2^k * binomial(2*n+k,k) * binomial(3*n-k,n-k).
%F A386896 a(n) = binomial(3*n, n)*hypergeom([-1-5*n, -n], [-3*n], -1). - _Stefano Spezia_, Aug 07 2025
%F A386896 D-finite with recurrence 202*n*(n-1)*(2*n-1)*(2*n-3)*a(n) -3*(n-1)*(2*n-3) *(14093*n^2-15245*n+5226)*a(n-1) +4*(355081*n^4 -1597876*n^3 +2789549*n^2 -2405270*n+926160)*a(n-2) -3840*(5*n-11)*(5*n-9) *(5*n-13)*(5*n-12)*a(n-3)=0. - _R. J. Mathar_, Aug 21 2025
%o A386896 (PARI) a(n) = sum(k=0, n, binomial(5*n+1, k)*binomial(3*n-k, n-k));
%Y A386896 Cf. A386812, A386895, A386897, A386898.
%K A386896 nonn
%O A386896 0,2
%A A386896 _Seiichi Manyama_, Aug 07 2025