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 A056939 Number of antichains (or order ideals) in the poset 3*m*n or plane partitions with rows <= m, columns <= n and entries <= 3 17
 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; text; internal format)
 OFFSET 0,5 COMMENTS Determinants of 3 X 3 subarrays of Pascal's triangle A007318 (a matrix entry being set to 0 when not present). - Gerald McGarvey, Feb 24 2005 Also determinants of 3x3 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+1),C(n,k),C(n,k-1); C(n,k+2),C(n,k+1),C(n,k)]. - Peter Bala, May 10 2012 Contribution from Gary W. Adamson, Jul 10 2012: (Start) The triangular view of this triangle is 1; 1,1; 1,4,1; 1,10,10,1; 1,20,50,20,1; The n-th row of this triangle is generated by applying the ConvOffs transform to the first n terms of 1, 4, 10, 20,.. (A000292 without leading zero). See A214281 for a procedural definition of the transformation and search "ConvOffs" for more examples. (End) REFERENCES Berman and Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), p. 103-124 R. P. Stanley, Theory and application of plane partitions. II. Studies in Appl. Math. 50 (1971), p. 259-279. Thm. 18.1 LINKS Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4. J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124. [Annotated scanned copy] P. A. MacMahon, Combinatory analysis, section 495, 1916 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, 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) EXAMPLE 1      1      1      1      1      1 ...      1      4     10     20     35     56 ...      1     10     50    175    490   1176 ...      1     20    175    980   4116  14112 ...      1     35    490   4116  24696 116424 ...      1     56   1176  14112 116424 731808 ... 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, Jan 28 2009 *) 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 KEYWORD nonn,easy,tabl,nice AUTHOR STATUS approved

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Last modified October 18 15:41 EDT 2019. Contains 328162 sequences. (Running on oeis4.)