OFFSET
0,2
COMMENTS
Second binomial transform of A002293.
FORMULA
a(n) = Sum_{k=0..n} binomial(n,k) * binomial(4*k,k) * 2^(n-k) / (3*k+1).
a(n) = 2^n*F([1/4, 1/2, 3/4, -n], [2/3, 1, 4/3], -2^7/3^3), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 22 2021
a(n) ~ 2^(n - 10) * 155^(n + 3/2) / (sqrt(Pi) * n^(3/2) * 3^(3*n + 3/2)). - Vaclav Kotesovec, Nov 26 2021
MATHEMATICA
nmax = 21; A[_] = 0; Do[A[x_] = 1/(1 - 2 x) + x (1 - 2 x)^2 A[x]^4 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n, k] Binomial[4 k, k] 2^(n - k)/(3 k + 1), {k, 0, n}], {n, 0, 21}]
PROG
(PARI) a(n) = sum(k=0, n, binomial(n, k)*binomial(4*k, k)*2^(n-k)/(3*k+1)); \\ Michel Marcus, Nov 23 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Nov 22 2021
STATUS
approved