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a(n) = binomial(5*n, n)*hypergeom([-4*n, -n], [-5*n], -1).
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%I #23 Jan 09 2024 11:03:32

%S 1,9,145,2625,50049,982729,19665841,398796225,8166636545,168502295625,

%T 3497529199185,72949645000065,1527671538372225,32100078290806665,

%U 676451066002195825,14290577765009652865,302557549412667613185,6417968867896642617225,136371773642235542394385

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

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

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

%H Seiichi Manyama, <a href="/A331329/b331329.txt">Table of n, a(n) for n = 0..747</a>

%H Lin Yang, Yu-Yuan Zhang, and Sheng-Liang Yang, <a href="https://doi.org/10.1016/j.laa.2023.12.021">The halves of Delannoy matrix and Chung-Feller properties of the m-Schröder paths</a>, Linear Alg. Appl. (2024).

%F a(n) ~ sqrt(5 + 21/sqrt(17)) * (349 + 85*sqrt(17))^n / (sqrt(Pi*n) * 2^(5*n + 2)). - _Vaclav Kotesovec_, Feb 13 2021

%t a[n_] := Binomial[5 n, n] Hypergeometric2F1[-4 n, -n, -5 n, -1];

%t Array[a, 19, 0]

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

%Y Cf. A008288, A181675, A341476, A341477.

%K nonn

%O 0,2

%A _Peter Luschny_, Jan 31 2020