OFFSET
0,2
FORMULA
a(n) = [x^n] 1/((1-x^3) * (1-x)^(3*n)).
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
From Vaclav Kotesovec, Apr 08 2024: (Start)
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).
a(n) ~ 2^(8*n + 9/2) / (7 * sqrt(Pi*n) * 3^(3*n + 3/2)). (End)
PROG
(PARI) a(n) = sum(k=0, n\3, binomial(4*n-3*k-1, n-3*k));
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Apr 05 2024
STATUS
approved