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A371772 a(n) = Sum_{k=0..floor(n/3)} binomial(5*n-3*k-1,n-3*k).

%I A371772 #13 Apr 08 2024 18:48:57
%S A371772 1,4,36,365,3892,42714,477621,5411109,61901268,713435333,8271470666,
%T A371772 96361329024,1127086021461,13227336997645,155680966681101,
%U A371772 1836862248992565,21719923705450260,257316706385394615,3053599633736172765,36292098436808314572,431918050456887676362
%N A371772 a(n) = Sum_{k=0..floor(n/3)} binomial(5*n-3*k-1,n-3*k).
%F A371772 a(n) = [x^n] 1/((1-x^3) * (1-x)^(4*n)).
%F A371772 a(n) = binomial(5*n-1, n)*hypergeom([1, (1-n)/3, (2-n)/3, -n/3], [(1-5*n)/3, (2-5*n)/3, 1-5*n/3], 1). - _Stefano Spezia_, Apr 06 2024
%F A371772 From _Vaclav Kotesovec_, Apr 08 2024: (Start)
%F A371772 Recurrence: 72*n*(2*n - 1)*(4*n - 3)*(4*n - 1)*(899*n^2 - 2355*n + 1534)*a(n) = (25514519*n^6 - 117751221*n^5 + 212960873*n^4 - 191684487*n^3 + 89835824*n^2 - 20567076*n + 1769040)*a(n-1) - 5*(5*n - 7)*(5*n - 6)*(5*n - 4)*(5*n - 3)*(899*n^2 - 557*n + 78)*a(n-2).
%F A371772 a(n) ~ 5^(5*n + 5/2) / (31 * sqrt(Pi*n) * 2^(8*n + 3/2)). (End)
%o A371772 (PARI) a(n) = sum(k=0, n\3, binomial(5*n-3*k-1, n-3*k));
%Y A371772 Cf. A371758, A371770, A371771.
%Y A371772 Cf. A371756.
%K A371772 nonn
%O A371772 0,2
%A A371772 _Seiichi Manyama_, Apr 05 2024