OFFSET
0,2
COMMENTS
Partial sums of A002293.
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..1000
FORMULA
G.f. A(x) satisfies: A(x) = 1 / (1 - x) + x * (1 - x)^3 * A(x)^4.
a(n) ~ 2^(8*n + 17/2) / (229 * sqrt(Pi) * n^(3/2) * 3^(3*n + 3/2)). - Vaclav Kotesovec, Jul 28 2021
D-finite with recurrence 3*n*(3*n-1)*(3*n+1)*a(n) +(-283*n^3+384*n^2-173*n+24)*a(n-1) +8*(4*n-3)*(2*n-1)*(4*n-1)*a(n-2)=0. - R. J. Mathar, Aug 05 2021
MATHEMATICA
Table[Sum[Binomial[4 k, k]/(3 k + 1), {k, 0, n}], {n, 0, 22}]
nmax = 22; A[_] = 0; Do[A[x_] = 1/(1 - x) + x (1 - x)^3 A[x]^4 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
PROG
(PARI) a(n) = sum(k=0, n, binomial(4*k, k)/(3*k+1)); \\ Michel Marcus, Jul 28 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Jul 28 2021
STATUS
approved