0,3

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 (psb(AT)maths.warwick.ac.uk), Mar 07 2003

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 (deutsch(AT)duke.poly.edu), Apr 20 2004

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 DELEHAM (kolotoko(AT)wanadoo.fr) Apr 15 2005

Terms in row n are the coefficients of the Legendre polynomial P(n,2x+1) with increasing powers of x.

Contribution from Peter Bala (pbala(AT)toucansurf.com), Oct 28 2008: (Start)

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.

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.

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)

J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 366.

S. Fomin and A. Zelevinsky, Cluster algebras I: Foundations, J. Amer. Math. Soc. 15 (2002) no. 2, 497-529

S. Fomin and A. Zelevinsky, Y-Systems and generalized associahedra, Ann. of Math. (2) 158 (2003), no. 3, 977-1018.

R. A. Sulanke, Objects counted by the central Delannoy numbers. J. Integer Seq. 6 (2003), no. 1, Article 03.1.5.

Gabor Hetyei. The Stirling polynomial of a simplicial complex. Discrete and Computational Geometry 35, Number 3, March 2006, pp 437-455. [From Peter Bala (pbala(AT)toucansurf.com), Oct 28 2008]

T. D. Noe, Rows n=0..100 of triangle, flattened

S. Fomin and A. Zelevinsky, Cluster algebras I: Foundations, J. Amer. Math. Soc. 15 (2002), no. 2, 497-529.

S. Fomin and A. Zelevinsky, Y-systems and generalized associahedra, Ann. of Math. (2) 158 (2003), no. 3, 977-1018.

R. A. Sulanke, Objects counted by the central Delannoy numbers., J. Integer Seq. 6 (2003), no. 1, Article 03.1.5.

V. Strehl, Recurrences and Legendre transform

F. Chapoton, Enumerative properties of generalized associahedra

F. Ardila, M. Beck, S. Hosten, J. Pfeifle and K. Seashore, Root polytopes and growth series of root lattices [From Peter Bala (pbala(AT)toucansurf.com), Oct 28 2008]

S. Fomin, N. Reading, Root systems and generalized associahedra, Lecture notes for IAS/Park-City 2004. [From Peter Bala (pbala(AT)toucansurf.com), Oct 28 2008]

G. Hetyei, Face enumeration using generalized binomial coefficients. This is the draft version of Hetyei's paper referenced above. [From Peter Bala (pbala(AT)toucansurf.com), Oct 28 2008]

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.

G.f.=G(t, z)=1/sqrt(1-2z-4tz+z^2). Row generating polynomials=P_n(1+2z), i.e. T(n, k)=[z^k]P_n(1+2z), where P_n are the Legendre polynomials. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 20 2004

Sum_{k>=0} T(n, k)*A000172(k) = Sum_{k>=0} T(n, k)^2 = A005259(n). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jun 08 2005 - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jun 08 2005

1; 1,2; 1,6,6; 1,12,30,20; 1,20,90,140,70; ...

with(orthopoly): seq([1, seq(coeff(expand(P(n, 1+2*z)), z^k), k=1..n)], n=0..9);

(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!)

Columns include A000012, A002378, A033487 on the left and A000984, A002457, A002544 on the right. Main diagonal is A006480. Row sums are A001850.

Cf. A008459.

Cf. A104684

A033282 (f-vectors type A associahedra), A108625, A080721 (f-vectors type D associahedra) [From Peter Bala (pbala(AT)toucansurf.com), Oct 28 2008]

Sequence in context: A133314 A049019 A046651 this_sequence A089231 A052296 A019538

Adjacent sequences: A063004 A063005 A063006 this_sequence A063008 A063009 A063010

nonn,tabl,nice,easy

Henry Bottomley (se16(AT)btinternet.com), Jul 02 2001

More terms from Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jun 08 2005