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a(n) = Sum_{k=0..floor(n/3)} binomial(5*n-2*k,n-3*k).
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%I #12 Apr 08 2024 04:53:49

%S 1,5,45,456,4863,53383,597052,6765471,77407257,892270250,10346070471,

%T 120542238796,1410040212166,16549315766244,194792566133507,

%U 2298472850258746,27179673132135409,322013956853586970,3821532498419234994,45420775578132979989

%N a(n) = Sum_{k=0..floor(n/3)} binomial(5*n-2*k,n-3*k).

%F a(n) = [x^n] 1/((1-x-x^3) * (1-x)^(4*n)).

%F a(n) ~ 5^(5*n + 5/2) / (99 * sqrt(Pi*n) * 2^(8*n - 1/2)). - _Vaclav Kotesovec_, Apr 05 2024

%F a(n) = binomial(5*n, n)*hypergeom([1, (1-n)/3, (2-n)/3, -n/3], [(1-5*n)/2, -5*n/2, 1+4*n], -27/4). - _Stefano Spezia_, Apr 06 2024

%o (PARI) a(n) = sum(k=0, n\3, binomial(5*n-2*k, n-3*k));

%Y Cf. A144904, A371754, A371755.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Apr 05 2024