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%I #205 Apr 19 2024 11:29:59
%S 1,1,2,1,6,6,1,12,30,20,1,20,90,140,70,1,30,210,560,630,252,1,42,420,
%T 1680,3150,2772,924,1,56,756,4200,11550,16632,12012,3432,1,72,1260,
%U 9240,34650,72072,84084,51480,12870,1,90,1980,18480,90090,252252,420420,411840,218790,48620
%N T(n,k) = binomial(n,k)*binomial(n+k,k), 0 <= k <= n, triangle read by rows.
%C T(n,k) is the number of compatible k-sets of cluster variables in Fomin and Zelevinsky's Cluster algebra of finite type B_n. Take a row of this triangle regarded as a polynomial in x and rewrite as a polynomial in y := x+1. The coefficients of the polynomial in y give a row of triangle A008459 (squares of binomial coefficients). For example, x^2+6*x+6 = y^2+4*y+1. - _Paul Boddington_, Mar 07 2003
%C T(n,k) is the number of lattice paths from (0,0) to (n,n) using steps E=(1,0), N=(0,1) and D=(1,1) (i.e., bilateral Schroeder paths), having k N=(0,1) steps. E.g. T(2,0)=1 because we have DD; T(2,1) = 6 because we have NED, NDE, EDN, END, DEN and DNE; T(2,2)=6 because we have NNEE, NENE, NEEN, EENN, ENEN and ENNE. - _Emeric Deutsch_, Apr 20 2004
%C Another version of [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...] DELTA [0, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...] = 1; 1, 0; 1, 2, 0; 1, 6, 6, 0; 1, 12, 30, 20, 0; ..., where DELTA is the operator defined in A084938. - _Philippe Deléham_ Apr 15 2005
%C Terms in row n are the coefficients of the Legendre polynomial P(n,2x+1) with increasing powers of x.
%C From _Peter Bala_, Oct 28 2008: (Start)
%C Row n of this triangle is the f-vector of the simplicial complex dual to an associahedron of type B_n (a cyclohedron) [Fomin & Reading, p.60]. See A008459 for the corresponding h-vectors for associahedra of type B_n and A001263 and A033282 respectively for the h-vectors and f-vectors for associahedra of type A_n.
%C An alternative description of this triangle in terms of f-vectors is as follows. Let A_n be the root lattice generated as a monoid by {e_i - e_j: 0 <= i,j <= n+1}. Let P(A_n) be the polytope formed by the convex hull of this generating set. Then the rows of this array are the f-vectors of a unimodular triangulation of P(A_n) [Ardila et al.]. A008459 is the corresponding array of h-vectors for these type A_n polytopes. See A127674 (without the signs) for the array of f-vectors for type C_n polytopes and A108556 for the array of f-vectors associated with type D_n polytopes.
%C The S-transform on the ring of polynomials is the linear transformation of polynomials that is defined on the basis monomials x^k by S(x^k) = binomial(x,k) = x(x-1)...(x-k+1)/k!. Let P_n(x) denote the S-transform of the n-th row polynomial of this array. In the notation of [Hetyei] these are the Stirling polynomials of the type B associahedra. The first few values are P_1(x) = 2*x + 1, P_2(x) = 3*x^2 + 3*x + 1 and P_3(x) = (10*x^3 + 15*x^2 + 11*x + 3)/3. These polynomials have their zeros on the vertical line Re x = -1/2 in the complex plane, that is, the polynomials P_n(-x) satisfy a Riemann hypothesis. See A142995 for further details. The sequence of values P_n(k) for k = 0,1,2,3, ... produces the n-th row of A108625. (End)
%C This is the row reversed version of triangle A104684. - _Wolfdieter Lang_, Sep 12 2016
%C T(n, k) is also the number of (n-k)-dimensional faces of a convex n-dimensional Lipschitz polytope of real functions f defined on the set X = {1, 2, ..., n+1} which satisfy the condition f(n+1) = 0 (see Gordon and Petrov). - _Stefano Spezia_, Sep 25 2021
%C The rows seem to give (up to sign) the coefficients in the expansion of the integer-valued polynomial ((x+1)*(x+2)*(x+3)*...*(x+n) / n!)^2 in the basis made of the binomial(x+i,i). - _F. Chapoton_, Oct 09 2022
%C Chapoton's observation above is correct: the precise expansion is ((x+1)*(x+2)*(x+3)*...*(x+n)/ n!)^2 = Sum_{k = 0..n} (-1)^k*T(n,n-k)*binomial(x+2*n-k, 2*n-k), as can be verified using the WZ algorithm. For example, n = 3 gives ((x+1)*(x+2)*(x+3)/3!)^2 = 20*binomial(x+6,6) - 30*binomial(x+5,5) + 12*binomial(x+4,4) - binomial(x+3,3). - _Peter Bala_, Jun 24 2023
%D J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 366.
%D J. Ser, Les Calculs Formels des Séries de Factorielles. Gauthier-Villars, Paris, 1933, Table I, p. 92.
%D D. Zagier, Integral solutions of Apery-like recurrence equations, in: Groups and Symmetries: from Neolithic Scots to John McKay, CRM Proc. Lecture Notes 47, Amer. Math. Soc., Providence, RI, 2009, pp. 349-366.
%H T. D. Noe, <a href="/A063007/b063007.txt">Rows n = 0..100 of triangle, flattened</a>
%H F. Ardila, M. Beck, S. Hosten, J. Pfeifle and K. Seashore, <a href="http://arxiv.org/abs/0809.5123">Root polytopes and growth series of root lattices</a>, arXiv:0809.5123 [math.CO], 2008.
%H Cyril Banderier, <a href="http://lipn.univ-paris13.fr/~banderier/Papers/these.ps">Combinatoire analytique des chemins et des cartes</a>, Thesis (2001), page 49.
%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Barry3/barry93.html">Continued fractions and transformations of integer sequences</a>, JIS 12 (2009), #09.7.6.
%H H. J. Brothers, <a href="http://www.brotherstechnology.com/math/pascals-prism.html">Pascal's Prism: Supplementary Material</a>.
%H David Callan, <a href="https://arxiv.org/abs/2202.04649">A bijection for Delannoy paths</a>, arXiv:2202.04649 [math.CO], 2022.
%H F. Chapoton, <a href="http://www.mat.univie.ac.at/~slc/wpapers/s51chapoton.html">Enumerative properties of generalized associahedra</a>, Séminaire Lotharingien de Combinatoire, B51b (2004), 16 pp.
%H Johann Cigler, <a href="http://arxiv.org/abs/1501.04750">Some remarks and conjectures related to lattice paths in strips along the x-axis</a>, arXiv:1501.04750 [math.CO], 2015-2016.
%H Mark Dukes and Chris D. White, <a href="http://arxiv.org/abs/1603.01589">Web Matrices: Structural Properties and Generating Combinatorial Identities</a>, arXiv:1603.01589 [math.CO], 2016.
%H Mark Dukes and Chris D. White, <a href="https://doi.org/10.37236/5631">Web Matrices: Structural Properties and Generating Combinatorial Identities</a>, Electronic Journal Of Combinatorics, 23(1) (2016), #P1.45.
%H S. Fomin and N. Reading, <a href="https://arxiv.org/abs/math/0505518">Root systems and generalized associahedra</a>, Lecture notes for IAS/Park-City 2004; arXiv:math/0505518 [math.CO], 2005-2008.
%H S. Fomin and A. Zelevinsky, <a href="https://doi.org/10.1090/S0894-0347-01-00385-X">Cluster algebras I: Foundations</a>, J. Amer. Math. Soc. 15(2) (2002), 497-529.
%H S. Fomin and A. Zelevinsky, <a href="http://www.jstor.org/stable/3597238">Y-systems and generalized associahedra</a>, Ann. of Math. (2) 158 (2003), no. 3, 977-1018.
%H J. Gordon and F. Petrov, <a href="https://armj.math.stonybrook.edu/html-articles/Files/17-63/">Combinatorics of the Lipschitz Polytope</a>, Arnold Mathematical Journal (2016).
%H G. Hetyei, <a href="https://web.archive.org/web/20100703121455/http://math.uncc.edu//preprint/2004/2004_11.pdf">Face enumeration using generalized binomial coefficients</a>. This is the draft version of Hetyei's paper referenced below. [Archived version]
%H Gabor Hetyei, <a href="http://dx.doi.org/10.1007/s00454-005-1190-2">The Stirling polynomial of a simplicial complex</a> Discrete and Computational Geometry 35(3) (2006), 437-455.
%H Hsien-Kuei Hwang and Satoshi Kuriki, <a href="https://arxiv.org/abs/2404.06040">Integrated empirical measures and generalizations of classical goodness-of-fit statistics</a>, arXiv:2404.06040 [math.ST], 2024. See p. 11.
%H C. Lanczos, <a href="/A002457/a002457.pdf">Applied Analysis</a> (Annotated scans of selected pages). See page 514.
%H T. Manneville and V. Pilaud, <a href="http://arxiv.org/abs/1501.07152">Compatibility fans for graphical nested complexes</a>, arXiv preprint arXiv:1501.07152 [math.CO], 2015.
%H Thomas Selig, <a href="https://arxiv.org/abs/2202.06487">Combinatorial aspects of sandpile models on wheel and fan graphs</a>, arXiv:2202.06487 [math.CO], 2022.
%H J. Ser, <a href="/A002720/a002720_4.pdf">Les Calculs Formels des Séries de Factorielles</a>, Gauthier-Villars, Paris, 1933 [Local copy].
%H J. Ser, <a href="/A002720/a002720.pdf">Les Calculs Formels des Séries de Factorielles</a> (Annotated scans of some selected pages) See Table I, page 92.
%H V. Strehl, <a href="http://www.mat.univie.ac.at/~slc/opapers/s29strehl.html">Recurrences and Legendre transform</a>, Séminaire Lotharingien de Combinatoire, B29b (1992), 22 pp.
%H R. A. Sulanke, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL6/Sulanke/delannoy.html">Objects counted by the central Delannoy numbers.</a>, J. Integer Seq. 6 (2003), no. 1, Article 03.1.5.
%H D. Zagier, <a href="http://people.mpim-bonn.mpg.de/zagier/files/tex/AperylikeRecEqs/fulltext.pdf">Integral solutions of Apery-like recurrence equations</a>.
%F T(n, k) = (n+k)!/(k!^2*(n-k)!) = T(n-1, k)*(n+k)/(n-k) = T(n, k-1)*(n+k)*(n-k+1)/k^2 = T(n-1, k-1)*(n+k)*(n+k-1)/k^2.
%F binomial(x, n)^2 = Sum_{k>=0} T(n,k) * binomial(x, n+k). - _Michael Somos_, May 11 2012
%F T(n, k) = A109983(n, k+n). - _Michael Somos_, Sep 22 2013
%F G.f.: G(t, z) = 1/sqrt(1-2*z-4*t*z+z^2). Row generating polynomials = P_n(1+2z), i.e., T(n, k) = [z^k] P_n(1+2*z), where P_n are the Legendre polynomials. - _Emeric Deutsch_, Apr 20 2004
%F Sum_{k>=0} T(n, k)*A000172(k) = Sum_{k>=0} T(n, k)^2 = A005259(n). - _Philippe Deléham_, Jun 08 2005
%F 1 + z*d/dz(log(G(t,z)) = 1 + (1 + 2*t)*z + (1 + 8*t + 8*t^2)*z^2 + ... is the o.g.f. for a signed version of A127674. - _Peter Bala_, Sep 02 2015
%F If R(n,t) denotes the n-th row polynomial then x^3 * exp( Sum_{n >= 1} R(n,t)*x^n/n ) = x^3 + (1 + 2*t)*x^4 + (1 + 5*t + 5*t^2)*x^5 + (1 + 9*t + 21*t^2 + 14*t^3)*x^6 + ... is an o.g.f for A033282. - _Peter Bala_, Oct 19 2015
%F P(n,x) := 1/(1 + x)*Integral_{t = 0..x} R(n,t) dt are (modulo differences of offset) the row polynomials of A033282. - _Peter Bala_, Jun 23 2016
%F From _Peter Bala_, Mar 09 2018: (Start)
%F R(n,x) = Sum_{k = 0..n} binomial(2*k,k)*binomial(n+k,n-k)*x^k.
%F R(n,x) = Sum_{k = 0..n} binomial(n,k)^2*x^k*(1 + x)^(n-k).
%F n*R(n,x) = (1 + 2*x)*(2*n - 1)*R(n-1,x) - (n - 1)*R(n-2,x).
%F R(n,x) = (-1)^n*R(n,-1 - x).
%F R(n,x) = 1/n! * (d/dx)^n ((x^2 + x)^n). (End)
%F The row polynomials are R(n,x) = hypergeom([-n, n + 1], [1], -x). - _Peter Luschny_, Mar 09 2018
%F T(n,k) = C(n+1,k)*A009766(n,k). - _Bob Selcoe_, Jan 18 2020 (Connects this triangle with the Catalan triangle. - _N. J. A. Sloane_, Jan 18 2020)
%F If we let A(n,k) = (-1)^(n+k)*(2*k+1)*(n*(n-1)*...*(n-(k-1)))/((n+1)*...*(n+(k+1))) for n >= 0 and k = 0..n, and we consider both T(n,k) and A(n,k) as infinite lower triangular arrays, then they are inverses of one another. (Empty products are by definition 1.) See the example below. The rational numbers |A(n,k)| appear in Table II on p. 92 in Ser's (1933) book. - _Petros Hadjicostas_, Jul 11 2020
%F From _Peter Bala_, Nov 28 2021: (Start)
%F Row polynomial R(n,x) = Sum_{k >= n} binomial(k,n)^2 * x^(k-n)/(1+x)^(k+1) for x > -1/2.
%F R(n,x) = 1/(1 + x)^(n+1) * hypergeom([n+1, n+1], [1], x/(1 + x)).
%F R(n,x) = (1 + x)^n * hypergeom([-n, -n], [1], x/(1 + x)).
%F R(n,x) = hypergeom([(n+1)/2, -n/2], [1], -4*x*(1 + x)).
%F If we set R(-1,x) = 1, we can run the recurrence n*R(n,x) = (1 + 2*x)*(2*n - 1)*R(n-1,x) - (n - 1)*R(n-2,x) backwards to give R(-n,x) = R(n-1,x).
%F R(n,x) = [t^n] ( (1 + t)*(1 + x*(1 + t)) )^n. (End)
%F n*T(n,k) = (2*n-1)*T(n-1,k) + (4*n-2)*T(n-1,k-1) - (n-1)*T(n-2,k). - _Fabián Pereyra_, Jun 30 2022
%e The triangle T(n, k) starts:
%e n\k 0 1 2 3 4 5 6 7
%e 0: 1
%e 1: 1 2
%e 2: 1 6 6
%e 3: 1 12 30 20
%e 4: 1 20 90 140 70
%e 5: 1 30 210 560 630 252
%e 6: 1 42 420 1680 3150 2772 924
%e 7: 1 56 756 4200 11550 16632 12012 3432
%e row n = 8: 1 72 1260 9240 34650 72072 84084 51480 12870,
%e row n = 9: 1 90 1980 18480 90090 252252 420420 411840 218790 48620,
%e row n = 10: 1 110 2970 34320 210210 756756 1681680 2333760 1969110 923780 184756.
%e ... reformatted by _Wolfdieter Lang_, Sep 12 2016
%e From _Petros Hadjicostas_, Jul 11 2020: (Start)
%e Its inverse (from Table II, p. 92, in Ser's book) is
%e 1;
%e -1/2, 1/2;
%e 1/3, -1/2, 1/6;
%e -1/4, 9/20, -1/4, 1/20;
%e 1/5, -2/5, 2/7, -1/10, 1/70;
%e -1/6, 5/14, -25/84, 5/36, -1/28, 1/252;
%e 1/7, -9/28, 25/84, -1/6, 9/154, -1/84, 1/924;
%e ... (End)
%p p := (n,x) -> orthopoly[P](n,1+2*x): seq(seq(coeff(p(n,x),x,k), k=0..n), n=0..9);
%t Flatten[Table[Binomial[n, k]Binomial[n + k, k], {n, 0, 10}, {k, 0, n}]] (* _Harvey P. Dale_, Dec 24 2011 *)
%t Table[CoefficientList[Hypergeometric2F1[-n, n + 1, 1, -x], x], {n, 0, 9}] // Flatten
%t (* _Peter Luschny_, Mar 09 2018 *)
%o (PARI) {T(n, k) = local(t); if( n<0, 0, t = (x + x^2)^n; for( k=1, n, t=t'); polcoeff(t, k) / n!)} /* _Michael Somos_, Dec 19 2002 */
%o (PARI) {T(n, k) = binomial(n, k) * binomial(n+k, k)} /* _Michael Somos_, Sep 22 2013 */
%o (PARI) {T(n, k) = if( k<0 || k>n, 0, (n+k)! / (k!^2 * (n-k)!))} /* _Michael Somos_, Sep 22 2013 */
%o (Haskell)
%o a063007 n k = a063007_tabl !! n !! k
%o a063007_row n = a063007_tabl !! n
%o a063007_tabl = zipWith (zipWith (*)) a007318_tabl a046899_tabl
%o -- _Reinhard Zumkeller_, Nov 18 2014
%o (Magma) /* As triangle: */ [[Binomial(n,k)*Binomial(n+k,k): k in [0..n]]: n in [0.. 15]]; // _Vincenzo Librandi_, Sep 03 2015
%Y See A331430 for an essentially identical triangle, except with signed entries.
%Y Columns include A000012, A002378, A033487 on the left and A000984, A002457, A002544 on the right.
%Y Main diagonal is A006480.
%Y Row sums are A001850. Alternating row sums are A033999.
%Y Cf. A007318, A008459, A009766, A046899, A104684, A109983, A127674.
%Y Cf. A033282 (f-vectors type A associahedra), A108625, A080721 (f-vectors type D associahedra).
%Y The Apéry-like numbers [or Apéry-like sequences, Apery-like numbers, Apery-like sequences] include A000172, A000984, A002893, A002895, A005258, A005259, A005260, A006077, A036917, A063007, A081085, A093388, A125143 (apart from signs), A143003, A143007, A143413, A143414, A143415, A143583, A183204, A214262, A219692,A226535, A227216, A227454, A229111 (apart from signs), A260667, A260832, A262177, A264541, A264542, A279619, A290575, A290576. (The term "Apery-like" is not well-defined.)
%K nonn,tabl,nice,easy,changed
%O 0,3
%A _Henry Bottomley_, Jul 02 2001
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