3,3

T(n+3,k) is also the number of compatible k-sets of cluster variables in Fomin and Zelevinsky's cluster algebra of finite type A_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 the triangle of Narayana numbers A001263. For example x^2+5*x+5=y^2+3*y+1. - Paul Boddington (psb(AT)maths.warwick.ac.uk), Mar 07 2003

Number of standard Young tableaux of shape (k+1,k+1,1^(n-k-3)), where 1^(n-k-3) denotes a sequence of n-k-3 1's (see the Stanley reference).

Number of k dimensional 'faces' of the n dimensional associahedron (see Simion, p. 168). - Mitch Harris (maharri(AT)gmail.com), Jan 16 2007

Mirror image of triangle A126216 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 19 2007

For relation to Lagrange inversion or series reversion and the geometry of associahedra or Stasheff polytopes (and other combinatorial objects) see A133437. [From Tom Copeland (tcjpn(AT)msn.com), Sep 29 2008]

Row generating polynomials 1/(n+1)*Jacobi_P(n,1,1,2*x+1). Row n of this triangle is the f-vector of the simplicial complex dual to an associahedron of type A_n [Fomin & Reading, p.60]. See A001263 for the corresponding array of h-vectors for associahedra of type A_n. See A063007 and A080721 for the f-vectors for associahedra of type B and type D respectively. [From Peter Bala (pbala(AT)toucansurf.com), Oct 28 2008]

D. Beckwith, Legendre polynomials and polygon dissections?, Amer. Math. Monthly, 105 (1998), 256-257.

A. Cayley, On the partitions of a polygon, Proc. London Math. Soc., 22 (1891), 237-262 = Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 13, pp. 93ff. (See p. 239.)

B. Drake, I. M. Gessel and G. Xin, Three proofs and a generalization of the Goulden-Litsyn-Shevelev conjecture ..., J. Integer Sequences, Vol. 10 (2007), #07.3.7.

P. Flajolet and M. Noy, Analytic combinatorics of non-crossing configurations, Discrete Math., 204, 1999, 203-229.

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.

G. Kreweras, Sur les partitions..., Discrete Math. 1 (1972), 333-350.

R. C. Read, On general dissections of a polygon, Aequat. Math. 18 (1978), 370-388.

R. Simion, "Convex Polytopes and Enumeration", Adv. in Appl. Math. 18 (1997) pp. 149-180.

R. P. Stanley, Polygon dissections and standard Young tableaux, J. Comb. Theory, Ser. A, 76, 175-177, 1996.

Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4. [From Tom Copeland (tcjpn(AT)msn.com), Nov 03 2008]

F. Chapoton, Enumerative properties of generalized associahedra

P. Flajolet and M. Noy, Analytic Combinatorics of Non-crossing Configurations, Discrete Math., 204, 1999, 203-229.

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. C. Read, On general dissections of a polygon, Aequat. Math. 18 (1978), 370-388.

R. P. Stanley, Polygon dissections and standard Young tableaux, J. Comb. Theory, Ser. A, 76, 175-177, 1996.

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.f. G=G(t, z) satisfies (1+t)G^2-z(1-z-2tz)G+tz^4=0.

T(n, k)=binomial(n-3, k)*binomial(n+k-1, k)/(k+1) for n >= 3, 0 <=k <=n-3.

Contribution from Tom Copeland (tcjpn(AT)msn.com), Nov 03 2008: (Start)

Two g.f.s (f1 and f2) for A033282 and their inverses (x1 and x2) can be derived from the Drake and Barry references.

1. a: f1(x,t) = y = {1 - (2t+1) x - sqrt[1 - (2t+1) 2x + x^2]}/[2x (t+1)]

= t x + (t + 2 t^2) x^2 + (t + 5 t^2 + 5 t^3) x^3 + ...

b: x1 = y/[t + (2t+1)y + (t+1)y^2] = y {1/[t/(t+1) + y] - 1/(1+y)}

= (y/t) - (1+2t)(y/t)^2 + (1+ 3t + 3t^2)(y/t)^3 +...

2. a: f2(x,t) = y = {1 - x - sqrt[(1-x)^2 - 4xt]}/[2(t+1)]

= (t/(t+1)) x + t x^2 + (t + 2 t^2) x^3 + (t + 5 t^2 + 5 t^3) x^4 + ...

b: x2 = y(t+1) [1- y(t+1)]/[t + y(t+1)]

= (t+1) (y/t) - (t+1)^3 (y/t)^2 + (t+1)^4 (y/t)^3 + ...

c: y/x2(y,t) = [t/(t+1) + y] / [1- y(t+1)]

= t/(t+1) + (1+t) y + (1+t)^2 y^2 + (1+t)^3 y^3 + ...

x2(y,t) can be used along with the Lagrange inversion for an o.g.f. (A133437)

to generate A033282 and show that A133437 is a refinement of A033282,

i.e., a refinement of the f-polynomials of the associahedra, the Stasheff polytopes.

y/x2(y,t) can be used along with the indirect Lagrange inversion (A134264)

to generate A033282 and show that A134264 is a refinement of A001263, i.e.,

a refinement of the h-polynomials of the associahedra.

f1[x,t](t+1) gives a generator for A088617.

f1[xt,1/t](t+1) gives a generator for A060693, with inverse y/[1 + t + (2+t) y + y^2].

f1[x(t-1),1/(t-1)]t gives a generator for A001263, with inverse y/[t + (1+t) y + y^2].

The unsigned coefficients of x1(y t,t) are A074909, reverse rows of A135278. (End)

G.f.: 1/(1-xy-(x+xy)/(1-xy/(1-(x+xy)/(1-xy/(1-(x+xy)/(1-xy/(1-.... (continued fraction). [From Paul Barry (pbarry(AT)wit.ie), Feb 06 2009]

1; 1,2; 1,5,5; 1,9,21,14; 1,14,56,84,42;

Diagonals : A000012, A000096, A033275, A033276, A033277, A033278, A033279; A000108, A002054, A002055, A002056, A007160, A033280, A033281 Row sums : A001003 (Schroeder numbers, first term omitted) . See A086810 for another version.

A007160 is a diagonal. Cf. A001263.

With leading zero: A086810.

Cf. A019538 'faces' of the permutohedron.

Cf. A063007 (f-vectors type B associahedra), A080721 (f-vectors type D associahedra), A126216 (mirror image). [From Peter Bala (pbala(AT)toucansurf.com), Oct 28 2008]

Sequence in context: A145882 A111785 A021468 this_sequence A126350 A079502 A126124

Adjacent sequences: A033279 A033280 A033281 this_sequence A033283 A033284 A033285

nonn,tabl,easy

N. J. A. Sloane (njas(AT)research.att.com).

Added a missing factor of 2 for expansions of f1 and f2 Tom Copeland (tcjpn(AT)msn.com), Apr 12 2009