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A048601
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Robbins triangle read by rows: T(n,k) = number of alternating sign n X n matrices with a 1 at top of column k (n >= 1, 1<=k<=n)
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10
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1, 1, 1, 2, 3, 2, 7, 14, 14, 7, 42, 105, 135, 105, 42, 429, 1287, 2002, 2002, 1287, 429, 7436, 26026, 47320, 56784, 47320, 26026, 7436, 218348, 873392, 1813968, 2519400, 2519400, 1813968, 873392, 218348, 10850216, 48825972, 113927268, 179028564
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OFFSET
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1,4
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COMMENTS
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An alternating sign matrix is a matrix of 0's and 1's such that (a) the sum of each row and column is 1; (b) the nonzero entries in each row and column alternate in sign.
Named after the American mathematician David Peter Robbins (1942-2003). - Amiram Eldar, Jun 13 2021
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REFERENCES
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David Bressoud, Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture, Cambridge University Press, 1999, p. 5.
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LINKS
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FORMULA
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T(n,k) = binomial(n+k-2, k-1)*((2*n-k-1)!/(n-k)!) * product(((3*j+1)!/(n+j)!), j=0..n-2);
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EXAMPLE
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Triangle begins:
1,
1, 1,
2, 3, 2,
7, 14, 14, 7,
42, 105, 135, 105, 42,
429, 1287, 2002, 2002, 1287, 429,
7436, 26026, 47320, 56784, 47320, 26026, 7436,
...
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MAPLE
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T:=(n, k)-> binomial(n+k-2, k-1)*((2*n-k-1)!/(n-k)!)*product(((3*j+1)!/(n+j)!), j=0..n-2);
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MATHEMATICA
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t[n_, k_] := Binomial[n+k-2, k-1]*((2*n-k-1)!/(n-k)!)*Product[((3*j+1)!/(n+j)!), {j, 0, n-2}]; Table[t[n, k], {n, 1, 9}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 12 2012, from formula *)
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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