A181675 -id:A181675 - cp's OEIS frontend

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

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

Offset: 0

Peter Luschny, Jan 31 2020

nonn

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

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

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).

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

Cf. A008288, A181675, A341476, A341477.

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

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

A341470 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..n} binomial(kn,n-j) binomial(k*n+j,j).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 13, 1, 1, 7, 41, 63, 1, 1, 9, 85, 377, 321, 1, 1, 11, 145, 1159, 3649, 1683, 1, 1, 13, 221, 2625, 16641, 36365, 8989, 1, 1, 15, 313, 4991, 50049, 246047, 369305, 48639, 1, 1, 17, 421, 8473, 118721, 982729, 3707509, 3800305, 265729, 1

Offset: 0

Seiichi Manyama, Feb 13 2021

			Square array begins:
  1,    1,     1,      1,      1,       1, ...
  1,    3,     5,      7,      9,      11, ...
  1,   13,    41,     85,    145,     221, ...
  1,   63,   377,   1159,   2625,    4991, ...
  1,  321,  3649,  16641,  50049,  118721, ...
  1, 1683, 36365, 246047, 982729, 2908411, ...

Eric Weisstein's World of Mathematics, Delannoy Number.

Columns k=0..5 give A000012, A001850, A026000, A026001, A331329, A341491.
Rows n=0..2 give A000012, A005408, A102083.
Main diagonal gives A181675(n+1).
Cf. A008288.

T(n, k) = sum(j=0, n, binomial(k*n, n-j)*binomial(k*n+j, j));

T(n, k) = sum(j=0, n, 2^j*binomial(n, j)*binomial(k*n, j));

T(n,k) = A008288(n,k*n).

T(n,k) = Sum_{j=0..n} 2^j * binomial(n,j) * binomial(k*n,j).

cp's OEIS Frontend

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

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A341470 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..n} binomial(kn,n-j) binomial(k*n+j,j).

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cp's OEIS Frontend

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

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Author

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Comments

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Programs

Mathematica

Formula

A341470 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..n} binomial(k*n,n-j) * binomial(k*n+j,j).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

PARI

Formula

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

A341470 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..n} binomial(kn,n-j) binomial(k*n+j,j).