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A386895 a(n) = Sum_{k=0..n} binomial(5*n+1,k) * binomial(2*n-k,n-k).

%I A386895 #17 Aug 21 2025 09:40:16
%S A386895 1,8,94,1220,16590,231808,3297154,47490696,690461070,10111370720,
%T A386895 148929775544,2203898519732,32741261744802,488010179737920,
%U A386895 7294326822378060,109294796958693520,1641111255497600910,24688289062391137056,372020649062760239080,5614219481885985162960
%N A386895 a(n) = Sum_{k=0..n} binomial(5*n+1,k) * binomial(2*n-k,n-k).
%F A386895 a(n) = [x^n] (1+x)^(5*n+1)/(1-x)^(n+1).
%F A386895 a(n) = [x^n] 1/((1-x)^(3*n+1) * (1-2*x)^(n+1)).
%F A386895 a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(5*n+1,k) * binomial(4*n-k,n-k).
%F A386895 a(n) = Sum_{k=0..n} 2^k * binomial(n+k,k) * binomial(4*n-k,n-k).
%F A386895 a(n) = binomial(2*n, n)*hypergeom([-1-5*n, -n], [-2*n], -1). - _Stefano Spezia_, Aug 07 2025
%F A386895 D-finite with recurrence +135*n*(n-1)*(3*n-1)*(3*n-2)*a(n) +3*(n-1)*(104049*n^3 -434754*n^2 +745789*n -439424)*a(n-1) +36*(517211*n^4 -4353801*n^3 +13137926*n^2 -17477238*n +8846684)*a(n-2) +16*(-11442763*n^4 +46270475*n^3 +85309279*n^2 -584322689*n +652846590)*a(n-3) -4585920*(5*n-16) *(5*n-14) *(5*n-18)*(5*n-17)*a(n-4)=0. - _R. J. Mathar_, Aug 21 2025
%o A386895 (PARI) a(n) = sum(k=0, n, binomial(5*n+1, k)*binomial(2*n-k, n-k));
%Y A386895 Cf. A386812, A386896, A386897, A386898.
%K A386895 nonn
%O A386895 0,2
%A A386895 _Seiichi Manyama_, Aug 07 2025