0,6

There are several versions of a Catalan triangle: see A009766, A008315, A028364.

Number of standard tableaux of shape (n-k,k) (0<=k<=floor(n/2)). Example: T(4,1)=3 because in th top row we can have 124, 134, or 123 (but not 234). - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 23 2004

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 796.

K. Fan, Structure of a Hecke algebra quotient, J. Amer. Math. Soc., 10 (1997), 139-167.

P. J. Larcombe, A question of proof..., Bull. Inst. Math. Applic. (IMA), 30, Nos. 3/4, 1994, 52-54.

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

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, December 1972 [alternative scanned copy].

R. K. Guy, Catwalks, Sandsteps and Pascal Pyramids, J. Integer Seqs., Vol. 3 (2000), #00.1.6

Index entries for sequences related to Chebyshev polynomials.

T(n, 0)=1 if n >= 0; T(2k, k)=T(2k-1, k-1) if k>0; T(n, k)=T(n-1, k-1)+T(n-1, k) if k=1, 2, ...[ n/2 ].

T(n, k) = C(n, k)-C(n, k-1) where C(n, k) is binomial coefficient.

Rows of Catalan triangle A008313 read backwards. Sum_{k>=0} T(n, k)^2 = A000108(n); A000108 : Catalan numbers. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 15 2004

Triangle begins:

1;

1;

1,1;

1,2;

1,3,2;

1,4,5;

1,5,9,5;

1,6,14,14;

1,7,20,28,14;

...

(PARI) T(n, k)=if(k<0|k>n\2, 0, if(n==0, 1, T(n-1, k-1) + T(n-1, k))).

T(2n, n) = A000108 (Catalan numbers), row sums = A001405 (central binomial coefficients).

This is also the positive half of the triangle in A008482 - Michael Somos

This is another reading (by shallow diagonals) of the triangle A009766, i.e. A008315[n] = A009766[A056536[n]].

Sequence in context: A062001 A049280 A108786 this_sequence A029635 A104741 A089353

Adjacent sequences: A008312 A008313 A008314 this_sequence A008316 A008317 A008318

nonn,tabf,nice,easy

njas

Expanded description from Clark Kimberling Jun 15 1997