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A056939
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Array read by antidiagonals: 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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25
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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
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OFFSET
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0,5
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COMMENTS
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Triangle of generalized binomial coefficients (n,k)_3; cf. A342889. This array is the main subject of the long article by Felsner et al. (2011). - N. J. A. Sloane, Apr 03 2021
This triangle is mentioned by Hoggatt (1977). - N. J. A. Sloane, Mar 27 2021
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 3 X 3 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
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)
Define polynomials p(n, x) = hypergeom([-1 - n, -n, 1 - n], [2, 3], -x). If the triangle is extended by the diagonal 1, 0, 0,... on the right side the resulting (0, 0)-based triangle is T*(n, k) = [x^k] p(n, x). The polynomials evaluated at x = 1 gives the number of Baxter permutations of length n (see the formula given by Richard L. Ollerton in A001181). - Peter Luschny, Dec 28 2022
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REFERENCES
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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
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LINKS
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FORMULA
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Product_{k=0..2} binomial(n+m+k, m+k)/binomial(n+k, k) 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)). - Roger L. Bagula, Jan 28 2009
T(n,k) = 2/((n+1)*(n+2)*(n+3))*C(n+1,k)*C(n+2,k+2)*C(n+3,k+1);
T(n,k) = 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)
The column generating functions of the square array (starting at column 1) are 1/(1 - x)^4, (1 + 3*x + x^2)/(1 - x)^7, (1 + 10*x + 20*x^2 + 10*x^3 + x^4)/(1 - x)^10, ..., where the numerator polynomials are the row polynomials of A087647. See Barry p. 31. - Peter Bala, Oct 18 2023
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EXAMPLE
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The initial rows of the array are:
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 ...
...
Considered as a triangle, the initial rows are:
[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]
...
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MAPLE
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bb := (k, l) -> binomial(k+l, k)*binomial(k+l+1, k)*binomial(k+l+2, k)*2/((k+1)^2*(k+2));
for k from 0 to 8 do
lprint([seq(bb(k, l), l=0..8)]);
od:
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MATHEMATICA
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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 *)
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PROG
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C=binomial;
T(n, k)=if(n==0&&k==0, 1, (C(n+1, k-1)*C(n+1, k)*C(n+1, k+1))/(C(n+1, 1)*C(n+1, 2)));
for(n=1, 10, for(k=1, n, print1(T(n, k), ", ")); print()); \\ Joerg Arndt, Jan 04 2024
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CROSSREFS
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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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STATUS
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approved
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