OFFSET
0,2
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..500
FORMULA
a(n) = Sum_{k=0..n} binomial(n+3*k,4*k) * binomial(4*k,k) / (3*k+1).
a(n) = F([(1+n)/3, (2+n)/3, (3+n)/3, -n], [2/3, 1, 4/3], -1), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 14 2021
a(n) ~ sqrt(1 + 3*r) / (2^(13/6) * sqrt(3*Pi) * (1-r)^(1/6) * n^(3/2) * r^(n + 1/3)), where r = 0.0766602099042102089064087954661556186872273232742446843... is the smallest real root of the equation 3^3 * (1-r)^4 = 4^4 * r. - Vaclav Kotesovec, Nov 15 2021
MATHEMATICA
nmax = 20; A[_] = 0; Do[A[x_] = (1 + x A[x]^4)/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n + 3 k, 4 k] Binomial[4 k, k]/(3 k + 1), {k, 0, n}], {n, 0, 20}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Nov 14 2021
STATUS
approved