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.

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.

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

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

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