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%I #89 Jan 04 2024 03:45:40
%S 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 490,56,1,1,84,1176,4116,4116,1176,84,1,1,120,2520,14112,24696,14112,
%U 2520,120,1,1,165,4950,41580,116424,116424,41580,4950,165,1
%N 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.
%C 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
%C This triangle is mentioned by Hoggatt (1977). - _N. J. A. Sloane_, Mar 27 2021
%C 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
%C 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
%C From _Gary W. Adamson_, Jul 10 2012: (Start)
%C The triangular view of this triangle is
%C 1;
%C 1, 1;
%C 1, 4, 1;
%C 1, 10, 10, 1;
%C 1, 20, 50, 20, 1;
%C 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)
%C 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
%D Berman and Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), p. 103-124
%D R. P. Stanley, Theory and application of plane partitions. II. Studies in Appl. Math. 50 (1971), p. 259-279. Thm. 18.1
%H Paul Barry, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Barry/barry91.html">On Integer-Sequence-Based Constructions of Generalized Pascal Triangles</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
%H J. Berman and P. Koehler, <a href="/A006356/a006356.pdf">Cardinalities of finite distributive lattices</a>, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124. [Annotated scanned copy]
%H Johann Cigler, <a href="https://arxiv.org/abs/2103.01652">Pascal triangle, Hoggatt matrices, and analogous constructions</a>, arXiv:2103.01652 [math.CO], 2021.
%H Johann Cigler, <a href="https://www.researchgate.net/publication/349376205_Some_observations_about_Hoggatt_triangles">Some observations about Hoggatt triangles</a>, Universität Wien (Austria, 2021).
%H Johann Cigler, <a href="https://arxiv.org/abs/2202.07298">Some observations about Hankel determinants of the columns of Pascal triangle and related topics</a>, arXiv:2202.07298 [math.CO], 2022.
%H Stefan Felsner, Eric Fusy, Marc Noy, and David Orden, <a href="https://doi.org/10.1016/j.jcta.2010.03.017">Bijections for Baxter families and related objects</a>, J. Combin. Theory Ser. A, 118(3):993-1020, 2011.
%H V. E. Hoggatt, Jr., <a href="/A006542/a006542.pdf">Letter to N. J. A. Sloane, Apr 1977</a>
%H P. A. MacMahon, <a href="http://www.hti.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=ABU9009">Combinatory analysis</a>, section 495, 1916.
%F Product_{k=0..2} binomial(n+m+k, m+k)/binomial(n+k, k) gives the array as a square.
%F 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
%F From _Peter Bala_, Oct 13 2011: (Start)
%F T(n,k) = 2/((n+1)*(n+2)*(n+3))*C(n+1,k)*C(n+2,k+2)*C(n+3,k+1);
%F 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.
%F 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).
%F 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)
%F 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
%e The initial rows of the array are:
%e 1 1 1 1 1 1 ...
%e 1 4 10 20 35 56 ...
%e 1 10 50 175 490 1176 ...
%e 1 20 175 980 4116 14112 ...
%e 1 35 490 4116 24696 116424 ...
%e 1 56 1176 14112 116424 731808 ...
%e ...
%e Considered as a triangle, the initial rows are:
%e [1],
%e [1, 1],
%e [1, 4, 1],
%e [1, 10, 10, 1],
%e [1, 20, 50, 20, 1],
%e [1, 35, 175, 175, 35, 1],
%e [1, 56, 490, 980, 490, 56, 1],
%e [1, 84, 1176, 4116, 4116, 1176, 84, 1],
%e [1, 120, 2520, 14112, 24696, 14112, 2520, 120, 1],
%e [1, 165, 4950, 41580, 116424, 116424, 41580, 4950, 165, 1],
%e [1, 220, 9075, 108900, 457380, 731808, 457380, 108900, 9075, 220, 1]
%e ...
%p # To get initial terms of the array - _N. J. A. Sloane_, Apr 20 2021
%p bb := (k,l) -> binomial(k+l,k)*binomial(k+l+1,k)*binomial(k+l+2,k)*2/((k+1)^2*(k+2));
%p for k from 0 to 8 do
%p lprint([seq(bb(k,l),l=0..8)]);
%p od:
%t 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 *)
%o (PARI) \\ cf. A359363
%o C=binomial;
%o 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)));
%o for(n=1,10,for(k=1,n,print1(T(n,k),", "));print()); \\ _Joerg Arndt_, Jan 04 2024
%Y Cf. A000372, A056932, A001263, A056940, A056941.
%Y Antidiagonals sum to A001181 (Baxter permutations). Cf. A197208.
%Y 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.
%K nonn,easy,tabl,nice
%O 0,5
%A _Mitch Harris_
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