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A362174
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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.
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1
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1, 1, 6, 180, 28000, 23152500, 103507455744, 2532712433771520, 342315030877028352000, 257389071045194840814562500, 1082814493908215083601185600000000, 25605944807023092680403880661295843852288, 3416912747607221845915134383073991514372073062400
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OFFSET
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0,3
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COMMENTS
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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.
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LINKS
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FORMULA
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a(n) = Product_{k=1..n} binomial(n+k-1,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
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EXAMPLE
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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.
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MAPLE
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a:= n-> mul(binomial(n+k-1, n-1), k=1..n):
seq(a(n), n=0..15);
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MATHEMATICA
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a[n_] := Product[Binomial[n + k - 1, n - 1], {k, 1, n}]
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PROG
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(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
(PARI) a(n) = prod(k=1, n, binomial(n+k-1, n-1)); \\ Michel Marcus, Jun 25 2023
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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