A341491 - OEIS

A341491

a(n) = binomial(6*n, n) * hypergeom([-5*n, -n], [-6*n], -1).

1, 11, 221, 4991, 118721, 2908411, 72616013, 1837271615, 46943003137, 1208483403179, 31297149356221, 814471993937855, 21281058718583873, 557930580979801755, 14669716953106628781, 386675596518995000191, 10214494658006725840897, 270345191656309313382475

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OFFSET

0,2

COMMENTS

In general, for k > 1, binomial(k*n, n) * hypergeom([(1-k)*n, -n], [-k*n], -1) ~ sqrt((k + (k^2 - k + 1) / sqrt(k^2 - 2*k + 2)) / (4*(k-1)*Pi*n)) * ((A341476(k) + A341477(k)*sqrt((k-1)^2 + 1)) / (k-1)^(k-1))^n.

LINKS

Michael De Vlieger, Table of n, a(n) for n = 0..697

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((6 + 31/sqrt(26))/(20*Pi*n)) * (42671 + 8346*sqrt(26))^n / 5^(5*n).

From Seiichi Manyama, Sep 13 2025: (Start)

a(n) = [x^n] (1-x)^n/(1-2*x)^(5*n+1).

a(n) = Sum_{k=0..n} 2^k * binomial(n,k) * binomial(5*n,k).

a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(n,k) * binomial(5*n+k,k).

a(n) = Sum_{k=0..n} binomial(n,k) * binomial(5*n+k,n). (End)

MATHEMATICA

Table[Binomial[6*n, n] * Hypergeometric2F1[-5*n, -n, -6*n, -1], {n, 0, 20}]