0,5

This is another reading (by shallow diagonals) of the triangle A009766; rows of Catalan triangle A008315 read backwards. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 15 2004

"The Catalan triangle is formed in the same manner as Pascal's triangle, except that no number may appear on the left of the vertical bar." [Conway and Smith]

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.

J. H. Conway and D. A. Smith, On Quaternions and Octonions, A K Peters, Ltd., Natick, MA, 2003. See p. 60. MR1957212 (2004a:17002)

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

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].

Index entries for sequences related to Chebyshev polynomials.

Row n: C(n-1, [ n/2 ]-k)-C(n-1, [ n/2 ]-k-2), k=0, 1, ..., n.

Sum_{k>=0} T(n, k)^2 = A000108(n); A000108: numbers of Catalan . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 14 2004

.......|...1

.......|.......1

.......|...1.......1

.......|.......2.......1

.......|...2.......3.......1

.......|.......5.......4.......1

.......|...5.......9.......5.......1

.......|......14......14.......6.......1

.......|..14......28......20.......7.......1

.......|......42......48......27.......8.......1

(PARI) T(n, k)=if(k<0|2*k>n, 0, polcoeff((1-x)*(1+x)^n, n\2-k)) /* Michael Somos May 28 2005 */

Cf. A039598, A039599. A053121 is essentially the same triangle.

Row sums = A001405 (central binomial coefficients).

Sequence in context: A117704 A078032 A162453 this_sequence A111377 A014046 A128065

Adjacent sequences: A008310 A008311 A008312 this_sequence A008314 A008315 A008316

nonn,tabf,nice,easy

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

More terms from Clark Kimberling (ck6(AT)evansville.edu)