A346680 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(4*k,k) / (3*k + 1).

1, 0, 4, 18, 122, 847, 6237, 47583, 373149, 2989111, 24354777, 201214021, 1681719343, 14193619647, 120800146953, 1035593096367, 8934344395053, 77510878324671, 675799844685937, 5918354494345863, 52037647837001257, 459200394617540288, 4065477723321641932

Offset

0,3

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) / (283 * sqrt(Pi) * n^(3/2) * 3^(3*n + 3/2)). - Vaclav Kotesovec, Jul 30 2021

D-finite with recurrence 3*n*(3*n-1)*(3*n+1)*a(n) -(n-1)*(229*n^2-155*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[(-1)^(n - k) 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, (-1)^(n-k)*binomial(4*k, k)/(3*k + 1)); \\ Michel Marcus, Jul 29 2021

Crossrefs

Cf. A002293, A032357, A188678, A345367, A346664.

Cf. A346681, A346682, A346683, A346684.

Sequence in context: A367489 A084661 A112294 * A292500 A228065 A317377

Adjacent sequences: A346677 A346678 A346679 * A346681 A346682 A346683

Keyword

nonn

Author

Ilya Gutkovskiy, Jul 29 2021

Status

approved