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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. 1
 1, 56, 1176, 14112, 116424, 731808, 3737448, 16195608, 61408347, 208416208, 644195552, 1837984512, 4892876352, 12259074816, 29115302688, 65937597264, 143107211709, 298915373064, 603074875480, 1178943365600, 2239226847000, 4142127132000, 7477931097000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 REFERENCES Richard P. Stanley: Enumerative Combinatorics, vol. 2, p. 378. LINKS Table of n, a(n) for n=0..22. Grigory M., Number of matrices with weakly increasing rows and columns, MathStackExchange. W. F. Wheatley and James Ethridge (Proposers), Comment from Alan H. Rapoport, Problem 84, Missouri Journal of Mathematical Sciences, volume 8, #2, Spring 1996, pages 97-102. FORMULA (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)). (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 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 CROSSREFS Cf. A140902-A140943. Sequence in context: A219583 A341429 A002009 * A160346 A219371 A183587 Adjacent sequences: A140898 A140899 A140900 * A140902 A140903 A140904 KEYWORD nonn AUTHOR R. H. Hardin, Jul 05 2008 STATUS approved

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Last modified July 23 19:43 EDT 2024. Contains 374553 sequences. (Running on oeis4.)