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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

Table of n, a(n) for n=0..54.

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

Mitch Harris

STATUS

approved

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Last modified February 23 09:40 EST 2018. Contains 299509 sequences. (Running on oeis4.)