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

T(n,k) is the number of n-digit binary words (length n sequences on {0,1}) containing k 1's such that no initial segment of the sequence has more 1's than 0's. [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Jul 31 2009]

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, 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

T(n,k)=Binomial(n,k)*(n-2k+1)/(n-k+1) [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Jul 31 2009]

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;

...

T(5,2)=5 because there are 5 such sequences: {0, 0, 0, 1, 1}, {0, 0, 1, 0, 1}, {0, 0, 1, 1, 0}, {0, 1, 0, 0, 1}, {0, 1, 0, 1, 0} [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Jul 31 2009]

Table[Binomial[k, i]*(k - 2 i + 1)/(k - i + 1), {k, 0, 20}, {i, 0, Floor[k/2]}] // Grid [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Jul 31 2009]

(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: A165999 A049280 A108786 this_sequence A029635 A104741 A167237

Adjacent sequences: A008312 A008313 A008314 this_sequence A008316 A008317 A008318

nonn,tabf,nice,easy

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

Expanded description from Clark Kimberling Jun 15 1997