A362174 Number of n X n matrices with nonnegative integer entries such that the sum of the elements of each row is equal to the index of that row.

1, 1, 6, 180, 28000, 23152500, 103507455744, 2532712433771520, 342315030877028352000, 257389071045194840814562500, 1082814493908215083601185600000000, 25605944807023092680403880661295843852288, 3416912747607221845915134383073991514372073062400

Offset: 0

Author

Michael Richard, Jun 12 2023

Keywords

• nonn

Comments

Also the number of n X n matrices with nonnegative integer entries such that the sum of the elements of each column is equal to the index of that column.

Examples

			a(1) = 1 as the only 1 X 1 matrix that satisfies the constraints is [1].
a(2) = 6 as there are 2 2d-vectors within the constraints with components that sum to 1 and independently 3 2d-vectors within the constraints with components that sum to 2. They are as follows: [[0 1],[1 1]], [[0 1],[2 0]], [[0 1],[0 2]], [[1 0],[1 1]], [[1 0],[2 0]], [[1 0],[0 2]],
a(3) = 180 as there are 3 3d-vectors within the constraints with components that sum to 1, 6 that sum to 2, and 10 that sum to 3. 3*6*10 = 180.
		

Links

• Michael Richard, Table of n, a(n) for n = 0..52

Crossrefs

Cf. A001700, A306789, A361749.

Programs

Maple

a:= n-> mul(binomial(n+k-1, n-1), k=1..n):
seq(a(n), n=0..15);

Mathematica

a[n_] := Product[Binomial[n + k - 1, n - 1], {k, 1, n}]

PARI

a(n) = prod(k=1, n, binomial(n+k-1,n-1)); \\ Michel Marcus, Jun 25 2023

Python

from math import comb, prod
def a(n): return prod(comb(n+k, n-1) for k in range(n))

Python

from math import factorial
from functools import lru_cache
@lru_cache(maxsize=None)
def A362174(n): return A362174(n-1)*(2*n-1)*factorial(2*n-2)**2//n//factorial(n-1)**3//(n-1)**(n-1) if n else 1 # Chai Wah Wu, Jun 26 2023

Formula

a(n) = Product_{k=1..n} binomial(n+k-1,n-1).

a(n) = A001700(n-1)*A306789(n-1) for n >= 1.

a(n) = a(n-1)*(2n-1)*(2n-2)!^2/(n*(n-1)!^3*(n-1)^(n-1)). - Chai Wah Wu, Jun 26 2023

a(x) = x^x*G(2x+1)*(G(x+1)^(x-1)/G(x+2)^(x+1)) where G(x) is the Barnes G-function is a differentiable continuation of a(n) to the nonnegative reals. - Michael Richard, Jun 27 2023

a(n) ~ A * 2^(2*n^2 - n/2 - 7/12) / (Pi^((n+1)/2) * exp(n^2/2 - n + 1/6) * n^(n/2 + 5/12)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Nov 19 2023