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A056939
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Number of antichains (or order ideals) in the poset 3*m*n or plane partitions with rows <= m, columns <= n and entries <= 3
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16
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1, 1, 1, 1, 4, 1, 1, 10, 10, 1, 1, 20, 50, 20, 1, 1, 35, 175, 175, 35, 1, 1, 56, 490, 980, 490, 56, 1, 1, 84, 1176, 4116, 4116, 1176, 84, 1, 1, 120, 2520, 14112, 24696, 14112, 2520, 120, 1, 1, 165, 4950, 41580, 116424, 116424, 41580, 4950, 165, 1
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| Determinants of 3 X 3 subarrays of Pascal's triangle A007318 (a matrix entry being set to 0 when not present). - Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), Feb 24 2005
Row sums are {1, 2, 6, 22, 92, 422, 2074, 10754, 58202, 326240, 1882960,...}. - Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jan 28 2009)
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REFERENCES
| Berman and Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), p. 103-124
P. A. MacMahon, Combinatory Analysis, section 495, 1916.
R. P. Stanley, Theory and application of plane partitions. II. Studies in Appl. Math. 50 (1971), p. 259-279. Thm. 18.1
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
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FORMULA
| Product[ C(n+m+k, m+k)/C(n+k, k), {k, 0, 2} ] gives the array as a square.
T(n,m)=2*Binomial[n, m]*Binomial[n + 1, m + 1]*Binomial[n + 2, m + 2]/(( n - m + 1)^2*(n - m + 2)) [From Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jan 28 2009]
From Peter Bala, Oct 13 2011: (Start)
T(n,k) = 2/((n+1)*(n+2)*(n+3))*C(n+1,k)*C(n+2,k+2)*C(n+3,k+1) =
2/((n+1)*(n+2)*(n+3))*C(n+1,k+1)*C(n+2,k)*C(n+3,k+2). Cf. A197208.
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 a(r,n) = n!*(n+1)!*...*(n+r)!. The triangle whose (n,k)-th entry is a(r,0)*a(r,n)/(a(r,k)*a(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
| Contribution from Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jan 28 2009: (Start)
{1},
{1, 1},
{1, 4, 1},
{1, 10, 10, 1},
{1, 20, 50, 20, 1},
{1, 35, 175, 175, 35, 1},
{1, 56, 490, 980, 490, 56, 1},
{1, 84, 1176, 4116, 4116, 1176, 84, 1},
{1, 120, 2520, 14112, 24696, 14112, 2520, 120, 1},
{1, 165, 4950, 41580, 116424, 116424, 41580, 4950, 165, 1},
{1, 220, 9075, 108900, 457380, 731808, 457380, 108900, 9075, 220, 1} (End)
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MATHEMATICA
| t[n_, m_] = 2*Binomial[n, m]*Binomial[n + 1, m + 1]* Binomial[n + 2, m + 2]/((n - m + 1)^2*(n - m + 2)); Flatten[Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}]] (* Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jan 28 2009 *)
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CROSSREFS
| Cf. A000372, A056932, A001263, A056940, A056941.
Antidiagonals sum to A001181 (Baxter permutations). A197208.
Sequence in context: A175124 A089447 A082680 * A202924 A142595 A174669
Adjacent sequences: A056936 A056937 A056938 * A056940 A056941 A056942
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KEYWORD
| nonn,easy,tabl,nice
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AUTHOR
| Mitch Harris (Harris.Mitchell(AT)mgh.harvard.edu)
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