Search: id:A056939
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%I A056939
%S A056939 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,
%T A056939 490,56,1,1,84,1176,4116,4116,1176,84,1,1,120,2520,14112,24696,14112,
%U A056939 2520,120,1,1,165,4950,41580,116424,116424,41580,4950,165,1
%N A056939 Number of antichains (or order ideals) in the poset 3*m*n or plane partitions 
               with rows <= m, columns <= n and entries <= 3
%C A056939 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
%C A056939 Row sums are {1, 2, 6, 22, 92, 422, 2074, 10754, 58202, 326240, 1882960,
               ...}. - Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jan 28 2009)
%D A056939 Berman and Koehler, Cardinalities of finite distributive lattices, Mitteilungen 
               aus dem Mathematischen Seminar Giessen, 121 (1976), p. 103-124
%D A056939 P. A. MacMahon, Combinatory Analysis, section 495, 1916.
%D A056939 R. P. Stanley, Theory and application of plane partitions. II. Studies 
               in Appl. Math. 50 (1971), p. 259-279. Thm. 18.1
%D A056939 Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal 
               Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
%F A056939 Product[ C(n+m+k, m+k)/C(n+k, k), {k, 0, 2} ]
%F A056939 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]
%e A056939 Contribution from Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jan 28 
               2009: (Start)
%e A056939 {1},
%e A056939 {1, 1},
%e A056939 {1, 4, 1},
%e A056939 {1, 10, 10, 1},
%e A056939 {1, 20, 50, 20, 1},
%e A056939 {1, 35, 175, 175, 35, 1},
%e A056939 {1, 56, 490, 980, 490, 56, 1},
%e A056939 {1, 84, 1176, 4116, 4116, 1176, 84, 1},
%e A056939 {1, 120, 2520, 14112, 24696, 14112, 2520, 120, 1},
%e A056939 {1, 165, 4950, 41580, 116424, 116424, 41580, 4950, 165, 1},
%e A056939 {1, 220, 9075, 108900, 457380, 731808, 457380, 108900, 9075, 220, 1} 
               (End)
%t A056939 Contribution from Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jan 28 
               2009: (Start)
%t A056939 t[n_, m_] = 2*Binomial[n, m]*Binomial[n + 1, m + 1]* Binomial[n + 2, 
               m + 2]/((n - m + 1)^2*(n - m + 2));
%t A056939 Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}]
%t A056939 Flatten[%] (End)
%Y A056939 Cf. A000372, A056932, A001263, A056940, A056941.
%Y A056939 Antidiagonals sum to A001181 (Baxter permutations)
%Y A056939 Sequence in context: A109955 A089447 A082680 this_sequence A142595 A140711 
               A164366
%Y A056939 Adjacent sequences: A056936 A056937 A056938 this_sequence A056940 A056941 
               A056942
%K A056939 nonn,easy,tabl,nice
%O A056939 0,5
%A A056939 Mitch Harris (Harris.Mitchell(AT)mgh.harvard.edu)

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