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A371771 a(n) = Sum_{k=0..floor(n/3)} binomial(4*n-3*k-1,n-3*k).

%I A371771 #21 Feb 28 2025 11:38:06
%S A371771 1,3,21,166,1377,11748,102088,898677,7987305,71517307,644134026,
%T A371771 5829345492,52964836184,482846377185,4414405051413,40458397722306,
%U A371771 371605426607673,3419639400458316,31521758873514301,291000881055737811,2690082750919841442
%N A371771 a(n) = Sum_{k=0..floor(n/3)} binomial(4*n-3*k-1,n-3*k).
%H A371771 Robert Israel, <a href="/A371771/b371771.txt">Table of n, a(n) for n = 0..1000</a>
%F A371771 a(n) = [x^n] 1/((1-x^3) * (1-x)^(3*n)).
%F A371771 a(n) = binomial(4*n-1, n)*hypergeom([1, (1-n)/3, (2-n)/3, -n/3], [(1-4*n)/3, 2*(1-2*n)/3, 1-4*n/3], 1). - _Stefano Spezia_, Apr 06 2024
%F A371771 From _Vaclav Kotesovec_, Apr 08 2024: (Start)
%F A371771 Recurrence: 81*n*(3*n - 2)*(3*n - 1)*(9037*n^4 - 61391*n^3 + 154035*n^2 - 169317*n + 68836)*a(n) = 27*(2394805*n^7 - 19820156*n^6 + 66654684*n^5 - 117198990*n^4 + 115250735*n^3 - 62650734*n^2 + 17209736*n - 1814400)*a(n-1) - 3*(7021749*n^7 - 58192764*n^6 + 196050236*n^5 - 345531070*n^4 + 340849311*n^3 - 186035886*n^2 + 51353864*n - 5443200)*a(n-2) + 8*(2*n - 3)*(4*n - 9)*(4*n - 7)*(9037*n^4 - 25243*n^3 + 24084*n^2 - 9272*n + 1200)*a(n-3).
%F A371771 a(n) ~ 2^(8*n + 9/2) / (7 * sqrt(Pi*n) * 3^(3*n + 3/2)). (End)
%p A371771 f:= proc(n) local k; add(binomial(4*n-3*k-1,n-3*k),k=0..n/3) end proc:
%p A371771 map(f, [$0..30]); # _Robert Israel_, Feb 28 2025
%o A371771 (PARI) a(n) = sum(k=0, n\3, binomial(4*n-3*k-1, n-3*k));
%Y A371771 Cf. A005810, A147855, A262977.
%Y A371771 Cf. A371758, A371770, A371772.
%K A371771 nonn
%O A371771 0,2
%A A371771 _Seiichi Manyama_, Apr 05 2024