A345367 a(n) = Sum_{k=0..n} binomial(4*k,k) / (3*k + 1).

1, 2, 6, 28, 168, 1137, 8221, 62041, 482773, 3845033, 31188921, 256757719, 2139691083, 18015030073, 153008796673, 1309402039993, 11279339531413, 97724562251137, 851035285261745, 7445189624293545, 65401191955640665, 576639234410182210, 5101317352349364430

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

Cf. A002293, A014137, A104859, A345368, A346065, A346646, A346671, A346672.

Sequence in context: A109570 A262002 A245633 * A156626 A207386 A088501

Adjacent sequences: A345364 A345365 A345366 * A345368 A345369 A345370

Keyword

nonn

Author

Ilya Gutkovskiy, Jul 28 2021

Status

approved