1,4

Triangle read by rows: T(n,k) = number of Dyck n-paths (A000108) containing k returns to ground level. E.g. the paths UDUUDD, UUDDUD each have 2 returns; so T(3,2)=2. Row sums over even-indexed columns are the Fine numbers A000957. - David Callan (callan(AT)stat.wisc.edu), Jul 25 2005

Triangular array of numbers a(n,k) = number of linear forests of k planted planar trees and n non-root nodes.

Catalan convolution triangle; with offset [0,0]: a(n,m)=(m+1)*binomial(2*n-m,n-m)/(n+1), n >= m >= 0, else 0. G.f. for column m: c(x)*(x*c(x))^m with c(x) g.f. for A000108 (Catalan). - Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Sep 12 2001

a(n+1,m+1), n >= m >= 0, a(n,m) := 0, n<m, has inverse matrix A030528(n,m)*(-1)^(n-m).

a(n,k)=number of Dyck paths of semilength n and having k returns to the axis. Also number of Dyck paths of semilength n and having first peak at height k. Also number of ordered trees with n edges and root degree k. Also number of ordered trees with n edges and having the leftmost leaf at level k. Also number of parallelogram polyominoes of semiperimeter n+1 and having k cells in the leftmost column. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 01 2004

Triangle T(n,k) with 1<=k<=n given by [0, 1, 1, 1, 1, 1, 1, 1, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...] = 1; 0, 1; 0, 1, 1; 0, 2, 2, 1; 0, 5, 5, 3, 1; 0, 14, 14, 9, 4, 1; ... where DELTA is the operator defined in A084938; essentially the same triangle as A059365 . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jun 14 2004

Number of Dyck paths of semilength and having k-1 peaks at height 2. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 31 2004

Riordan array (c(x),xc(x)), c(x) the g.f. of A000108. Inverse of Riordan array (1-x,x(1-x)). - Paul Barry (pbarry(AT)wit.ie), Jun 22 2005

Subtriangle of triangle A106566 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jan 07 2007

T(n, k) is also the number of order-preserving and order-decreasing full transformations (of an n-chain) with exactly k fixed points. [From A. Umar (aumarh(AT)squ.edu.om), Oct 02 2008]

S. Brlek, E. Duchi, E. Pergola and S. Rinaldi, On the equivalence problem for succession rules, Discr. Math., 298 (2005), 142-154.

E. Deutsch, Dyck path enumeration, Discrete Math., 204, 1999, 167-202.

W. Lang, On polynomials related to powers of the generating function of Catalan numbers, The Fibonacci Quart. 38 (2000) 408-19.

P. J. Larcombe and D. R. French, The Catalan number k-fold self-convolution identity: the original formulation, Journal of Combinatorial Mathematics and Combinatorial Computing 46 (2003) 191-204.

M. Aigner, Enumeration via ballot numbers, Discrete Math., 308 (2008), 2544-2563.

Higgins, Peter M. Combinatorial results for semigroups of order-preserving mappings. Math. Proc. Camb. Phil. Soc. 113 (1993), 281-296. [From A. Umar (aumarh(AT)squ.edu.om), Oct 02 2008]

J. L. Arregui, Tangent and Bernoulli numbers related to Motzkin and Catalan numbers by means of numerical triangles.

D. Callan, A recursive bijective approach to counting permutations...

N. T. Cameron, Random walks, trees and extensions of Riordan group techniques

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

D. Merlini, R. Sprugnoli and M. C. Verri, An algebra for proper generating trees

J. Noonan and D. Zeilberger, [math/9808080] The Enumeration of Permutations With a Prescribed Number of ``Forbidden'' Patterns

A. Reifegerste, On the diagram of 132-avoiding permutation.

A. Robertson, D. Saracino and D. Zeilberger, Refined restricted permutations.

J.-C. Novelli and J.-Y. Thibon, Noncommutative Symmetric Functions and Lagrange Inversion

Column k is the k-fold convolution of column 1. The triangle is also defined recursively by (i) entries outside triangle are 0, (ii) top left entry is 1, (iii) every other entry is sum of its east and northwest neighbor. - David Callan (callan(AT)stat.wisc.edu), Jul 25 2005

G.f.= txc/(1-txc), where c=(1-sqrt(1-4x))/(2x) is the g.f. of the Catalan numbers (A000108). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 01 2004

T(n,k) = C(2n-k, n-k)*(k+1)/(n+1). [From Paul D. Hanna (pauldhanna(AT)juno.com), Aug 11 2008]

Triangle begins

\ k..1....2....3....4....5....6

n\

1 |..1

2 |..1....1

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

4 |..5....5....3....1

5 |.14...14....9....4....1

6 |.42...42...28...14....5....1

7 |132..132...90...48...20....6....1

a := proc(n, k) if k<=n then k*binomial(2*n-k, n)/(2*n-k) else 0 fi end: seq(seq(a(n, k), k=1..n), n=1..10);

(PARI) T(n, k)=binomial(2*(n-k)+k, n-k)*(k+1)/(n+1) [From Paul D. Hanna (pauldhanna(AT)juno.com), Aug 11 2008]

Rows of Catalan triangle A009766 read backwards.

a(n, 1)= A000108(n-1). Row sums = A000108(n) (Catalan).

The following are all versions of (essentially) the same Catalan triangle: A009766, A030237, A033184, A059365, A099039, A106566, A130020, A047072.

Diagonals give A000108 A000245 A002057 A000344 A003517 A000588 A003518 A003519 A001392, ...

Cf. A116364 (row squared sums). [From Paul D. Hanna (pauldhanna(AT)juno.com), Aug 11 2008]

Sequence in context: A114292 A141751 A079222 this_sequence A110488 A134379 A108087

Adjacent sequences: A033181 A033182 A033183 this_sequence A033185 A033186 A033187

nonn,tabl

Christian G. Bower (bowerc(AT)usa.net)