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A140901 Number of 3 X 5 matrices with elements in 0..n with each row and each column in nondecreasing order. 3,5,n can be permuted, see formula.

%I A140901 #38 Jan 17 2025 09:27:57
%S A140901 1,56,1176,14112,116424,731808,3737448,16195608,61408347,208416208,
%T A140901 644195552,1837984512,4892876352,12259074816,29115302688,65937597264,
%U A140901 143107211709,298915373064,603074875480,1178943365600,2239226847000,4142127132000,7477931097000
%N A140901 Number of 3 X 5 matrices with elements in 0..n with each row and each column in nondecreasing order. 3,5,n can be permuted, see formula.
%D A140901 Richard P. Stanley: Enumerative Combinatorics, vol. 2,  p. 378.
%H A140901 Feihu Liu, Guoce Xin, and Chen Zhang, <a href="https://arxiv.org/abs/2412.18744">Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS</a>, arXiv:2412.18744 [math.CO], 2024. See p. 25.
%H A140901 Grigory M., <a href="https://math.stackexchange.com/q/643729">Number of matrices with weakly increasing rows and columns</a>, MathStackExchange.
%H A140901 W. F. Wheatley and James Ethridge (Proposers), Comment from Alan H. Rapoport, <a href="https://doi.org/10.35834/1996/0802092">Problem 84</a>, Missouri Journal of Mathematical Sciences, volume 8, #2, Spring 1996, pages 97-102.
%F A140901 (Empirical) Set p,q,r to n,5,3 (in any order) in s=p+q+r-1; a(n) = Product_{i=0..r-1} (binomial(s,p+i)*i!/(s-i)^(r-i-1)).
%F A140901 (Conjecture) G.f.: (1 + 40*x + 400*x^2 + 1456*x^3 + 2212*x^4 + 1456*x^5 + 400*x^6 + 40*x^7 + x^8)/(1-x)^16. - _Bruno Berselli_, May 08 2012
%F A140901 a(n) = Product_{i=1..3} Product_{j=1..5} Product_{k=1..n+1} (i + j + k - 1) / (i + j + k - 2). See the section on plane partitions with bounded part size in Stanley's reference. This comment is relevant to the sequences A140902 - A140943 as well. - _Sela Fried_, Oct 18 2023
%Y A140901 Cf. A140902-A140943.
%K A140901 nonn
%O A140901 0,2
%A A140901 _R. H. Hardin_, Jul 05 2008