A362174 - OEIS

%I #70 Nov 19 2023 06:34:07

%S 1,1,6,180,28000,23152500,103507455744,2532712433771520,

%T 342315030877028352000,257389071045194840814562500,

%U 1082814493908215083601185600000000,25605944807023092680403880661295843852288,3416912747607221845915134383073991514372073062400

%N 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.

%C 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.

%H Michael Richard, <a href="/A362174/b362174.txt">Table of n, a(n) for n = 0..52</a>

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

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

%F a(n) = a(n-1)*(2n-1)*(2n-2)!^2/(n*(n-1)!^3*(n-1)^(n-1)). - _Chai Wah Wu_, Jun 26 2023

%F 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

%F 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

%e a(1) = 1 as the only 1 X 1 matrix that satisfies the constraints is [1].

%e 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]],

%e 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.

%p a:= n-> mul(binomial(n+k-1, n-1), k=1..n):

%p seq(a(n), n=0..15);

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

%o (Python)

%o from math import comb, prod

%o def a(n): return prod(comb(n+k, n-1) for k in range(n))

%o (Python)

%o from math import factorial

%o from functools import lru_cache

%o @lru_cache(maxsize=None)

%o 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

%o (PARI) a(n) = prod(k=1, n, binomial(n+k-1,n-1)); \\ _Michel Marcus_, Jun 25 2023

%Y Cf. A001700, A306789, A361749.

%K nonn

%O 0,3

%A _Michael Richard_, Jun 12 2023