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A056940
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Number of antichains (or order ideals) in the poset 4*m*n or plane partitions with at most m rows and n columns and entries <= 4.
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17
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1, 1, 1, 1, 5, 1, 1, 15, 15, 1, 1, 35, 105, 35, 1, 1, 70, 490, 490, 70, 1, 1, 126, 1764, 4116, 1764, 126, 1, 1, 210, 5292, 24696, 24696, 5292, 210, 1, 1, 330, 13860, 116424, 232848, 116424, 13860, 330, 1, 1, 495, 32670, 457380, 1646568, 1646568, 457380, 32670, 495, 1
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OFFSET
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0,5
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COMMENTS
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Determinants of 4 X 4 subarrays of Pascal's triangle A007318 (a matrix entry being set to 0 when not present). - Gerald McGarvey, Feb 24 2005
Row sums are: {1, 2, 7, 32, 177, 1122, 7898, 60398, 494078, 4274228, 38763298, ...}. - Roger L. Bagula, Mar 08 2010
Also determinants of 4x4 arrays whose entries come from a single row: T(n,k) = det [C(n,k), C(n,k-1), C(n,k-2), C(n,k-3); C(n,k+1), C(n,k), C(n,k-1), C(n,k-2); C(n,k+2), C(n,k+1), C(n,k), C(n,k-1); C(n,k+3), C(n,k+2), C(n,k+1), C(n,k)]. - Peter Bala, May 10 2012
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LINKS
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FORMULA
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Product_{k=0..3} C(n+m+k, m+k)/C(n+k, k) gives the array as a square.
T(n,m,q) = c(n,q)/(c(m,q)*c(n-m,q)) with c(n,q) = Product_{i=1..n, j=0..q} (i + j), q = 3. - Roger L. Bagula, Mar 08 2010
T(n-1,k-1)*T(n,k+1)*T(n+1,k) = T(n-1,k)*T(n,k-1)*T(n+1,k+1).
Define f(r,n) = n!*(n+1)!*...*(n+r)!. The triangle whose (n,k)-th entry is f(r,0)*f(r,n)/(f(r,k)*f(r,n-k)) is A007318 (r = 0), A001263 (r = 1), A056939 (r = 2), A056940 (r = 3) and A056941 (r = 4). (End)
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EXAMPLE
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Triangle begins as:
1.
1, 1.
1, 5, 1.
1, 15, 15, 1.
1, 35, 105, 35, 1.
1, 70, 490, 490, 70, 1.
1, 126, 1764, 4116, 1764, 126, 1.
1, 210, 5292, 24696, 24696, 5292, 210, 1.
1, 330, 13860, 116424, 232848, 116424, 13860, 330, 1. - Roger L. Bagula, Mar 08 2010
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MATHEMATICA
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c[n_, q_] = Product[i + j, {j, 0, q}, {i, 1, n}];
T[n_, m_, q_] = c[n, q]/(c[m, q]*c[n - m, q]);
Table[T[n, k, 3], {n, 0, 10}, {k, 0, n}]//Flatten (* Roger L. Bagula, Mar 08 2010 *)(* modified by G. C. Greubel, Apr 13 2019 *)
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PROG
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(PARI) A056940(n, m)=prod(k=0, 3, binomial(n+m+k, m+k)/binomial(n+k, k)) \\ M. F. Hasler, Sep 26 2018
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CROSSREFS
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Antidiagonals sum to A005362 (Hoggatt sequence).
Triangles of generalized binomial coefficients (n,k)_m (or generalized Pascal triangles) for m = 1,...,12: A007318 (Pascal), A001263, A056939, A056940, A056941, A142465, A142467, A142468, A174109, A342889, A342890, A342891.
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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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