A331329 a(n) = binomial(5*n, n)*hypergeom([-4*n, -n], [-5*n], -1).

1, 9, 145, 2625, 50049, 982729, 19665841, 398796225, 8166636545, 168502295625, 3497529199185, 72949645000065, 1527671538372225, 32100078290806665, 676451066002195825, 14290577765009652865, 302557549412667613185, 6417968867896642617225, 136371773642235542394385

Offset

0,2

Comments

Special case of generalized Delannoy numbers (see cross-references):

T(n, k) = binomial(k*n, n)*hypergeom([(1-k)*n, -n], [-k*n], -1).

Links

Seiichi Manyama, Table of n, a(n) for n = 0..747

Lin Yang, Yu-Yuan Zhang, and Sheng-Liang Yang, The halves of Delannoy matrix and Chung-Feller properties of the m-Schröder paths, Linear Alg. Appl. (2024).

Formula

a(n) ~ sqrt(5 + 21/sqrt(17)) * (349 + 85*sqrt(17))^n / (sqrt(Pi*n) * 2^(5*n + 2)). - Vaclav Kotesovec, Feb 13 2021

From Seiichi Manyama, Sep 13 2025: (Start)

a(n) = [x^n] (1-x)^n/(1-2*x)^(4*n+1).

a(n) = Sum_{k=0..n} 2^k * binomial(n,k) * binomial(4*n,k).

a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(n,k) * binomial(4*n+k,k).

a(n) = Sum_{k=0..n} binomial(n,k) * binomial(4*n+k,n). (End)

Mathematica

a[n_] := Binomial[5 n, n] Hypergeometric2F1[-4 n, -n, -5 n, -1];
Array[a, 19, 0]

Crossrefs

Column k=4 of A341470.

Cf. A001850 (k=2), A026000 (k=3), A026001 (k=4), this sequence (k=5), A341491 (k=6).

Cf. A008288, A181675, A341476, A341477.

Sequence in context: A223371 A396159 A046529 * A388729 A064091 A132060

Adjacent sequences: A331326 A331327 A331328 * A331330 A331331 A331332

Keyword

nonn

Author

Peter Luschny, Jan 31 2020

Status

approved