0,2

T(n-2k-1,k) = number of n-th generation vertices in the tree of sequences with unit increase labeled by 2k+2 (cf. Zoran Sunik reference) . - Benoit Cloitre (benoit7848c(AT)orange.fr), Oct 07 2003

Number of lattice paths from (0,0) to (n,n) with steps E=(1,0) and N=(0,1) which touch but do not cross the line x-y=k+1 . - Herbert Kociemba (kociemba(AT)t-online.de), May 24 2004

Number of standard tableaux of shape (n+k+1, n-k-1) . - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 30 2004

Riordan array (c(x)^3,xc(x)^2) where c(x) is the g.f. of A000108. Inveres array is A109954. - Paul Barry (pbarry(AT)wit.ie), Jul 06 2005

V. E. Hoggatt, Jr. and M. Bicknell, Catalan and related sequences arising from inverses of Pascal's triangle matrices, Fib. Quart., 14 (1976), 395-405.

A. Papoulis, A new method of inversion of the Laplace transform, Quart. Applied Math.14 (1956), 405ff.

J. Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp., 29 (1975), 215-222.

Zoran Sunik, Self describing sequences and the Catalan family tree, Elect. J. Combin., 10 (No. 1, 2003).

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

Sum_{ k = 0, .., n-1} T(n, k) = binomial(2n, n-1) = A001791(n).

For the column k : expansion of x^(k+1)C^(2k+3) where C = (1-(1-4*x)^(1/2)/(2*x) is the g.f. of Catalan numbers A000108 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Feb 03 2004

T(n, k) = A039599(n, k+1) = A009766(n+k+1, n-k-1) = A033184(n+k+2, 2k+3) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), May 28 2005

Sum_{k>= 0} T(m, k)*T(n, k) = A000108(m+n) - A000108(m)*A000108(n). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), May 28 2005

T(n, k)=(2k+3)binomial(2n+2, n+k+2)/(n+k+3)=C(2n+2, n+k+2)-C(2n+2, n+k+3) [offset (0, 0)]. - Paul Barry (pbarry(AT)wit.ie), Jul 06 2005

1; 3, 1; 9, 5, 1; 28, 20, 7, 1; 90, 75, 35, 9, 1; 297, 275, 154, 54, 11, 1; . . .0

First columns : A000245, A000344, A000588, A001392, A000589, A000590.

See also : A000108, A001791(row sums), A050144.

Sequence in context: A091579 A005533 A112626 this_sequence A140714 A112932 A077895

Adjacent sequences: A050152 A050153 A050154 this_sequence A050156 A050157 A050158

nonn,tabl

Clark Kimberling (ck6(AT)evansville.edu)

Edited by Philippe DELEHAM, May 22 2005