A371771 - OEIS

%I #17 Apr 08 2024 18:48:46

%S 1,3,21,166,1377,11748,102088,898677,7987305,71517307,644134026,

%T 5829345492,52964836184,482846377185,4414405051413,40458397722306,

%U 371605426607673,3419639400458316,31521758873514301,291000881055737811,2690082750919841442

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

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

%F 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 From _Vaclav Kotesovec_, Apr 08 2024: (Start)

%F 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 a(n) ~ 2^(8*n + 9/2) / (7 * sqrt(Pi*n) * 3^(3*n + 3/2)). (End)

%o (PARI) a(n) = sum(k=0, n\3, binomial(4*n-3*k-1, n-3*k));

%Y Cf. A005810, A147855, A262977.

%Y Cf. A371758, A371770, A371772.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Apr 05 2024