A197208 -id:A197208 - cp's OEIS frontend

A056939 Array read by antidiagonals: number of antichains (or order ideals) in the poset 3mn or plane partitions with rows <= m, columns <= n and entries <= 3.

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

Offset: 0

Mitch Harris

Triangle of generalized binomial coefficients (n,k)A342889.%20This%20array%20is%20the%20main%20subject%20of%20the%20long%20article%20by%20Felsner%20et%20al.%20(2011).%20-%20_N.%20J.%20A.%20Sloane">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
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)
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

			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]
  ...

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

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]
Johann Cigler, Pascal triangle, Hoggatt matrices, and analogous constructions, arXiv:2103.01652 [math.CO], 2021.
Johann Cigler, Some observations about Hoggatt triangles, Universität Wien (Austria, 2021).
Johann Cigler, Some observations about Hankel determinants of the columns of Pascal triangle and related topics, arXiv:2202.07298 [math.CO], 2022.
Stefan Felsner, Eric Fusy, Marc Noy, and David Orden, Bijections for Baxter families and related objects, J. Combin. Theory Ser. A, 118(3):993-1020, 2011.
V. E. Hoggatt, Jr., Letter to N. J. A. Sloane, Apr 1977
P. A. MacMahon, Combinatory analysis, section 495, 1916.

Cf. A000372, A056932, A001263, A056940, A056941.
Antidiagonals sum to A001181 (Baxter permutations). Cf. A197208.
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.

# To get initial terms of the array - N. J. A. Sloane, Apr 20 2021
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:

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 *)

\\ cf. A359363
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

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
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);
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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A197209 Row sums of A197208.

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A056939 Array read by antidiagonals: number of antichains (or order ideals) in the poset 3mn or plane partitions with rows <= m, columns <= n and entries <= 3.

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cp's OEIS Frontend

A197209 Row sums of A197208.

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Author

Keywords

Links

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Programs

Mathematica

PARI

Formula

A056939 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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Author

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Programs

Maple

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PARI

Formula

A056939 Array read by antidiagonals: number of antichains (or order ideals) in the poset 3mn or plane partitions with rows <= m, columns <= n and entries <= 3.