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A131198 Triangle T(n,k), 0<=k<=n, read by rows, given by [1,0,1,0,1,0,1,0,...] DELTA [0,1,0,1,0,1,0,1,...] where DELTA is the operator defined in A084938 . 6
1, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 6, 6, 1, 0, 1, 10, 20, 10, 1, 0, 1, 15, 50, 50, 15, 1, 0, 1, 21, 105, 175, 105, 21, 1, 0, 1, 28, 196, 490, 490, 196, 28, 1, 0, 1, 36, 336, 1176, 1764, 1176, 336, 36, 1, 0, 1, 45, 540, 2520, 5292, 5292, 2520, 540, 45, 1, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

Mirror image of triangle A090181, another version of triangle of Narayana (A001263).

Equals A133336*A130595 as infinite lower triangular matrices . - Philippe Deléham, Oct 23 2007

LINKS

Table of n, a(n) for n=0..65.

P. Barry, Continued fractions and transformations of integer sequences, JIS 12 (2009) 09.7.6

P. Barry, On a Generalization of the Narayana Triangle, J. Int. Seq. 14 (2011) # 11.4.5

P. Barry, A. Hennessy, A Note on Narayana Triangles and Related Polynomials, Riordan Arrays, and MIMO Capacity Calculations , J. Int. Seq. 14 (2011) # 11.3.8

FindStat - Combinatorial Statistic Finder, The number of peaks of a Dyck path., The number of double rises of a Dyck path., The number of valleys of a Dyck path., The number of left oriented leafs except the first one of a binary tree., The number of left tunnels of a Dyck path.

Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.

FORMULA

Sum_{k, 0<=k<=n}T(n,k)*x^k = A000012(n), A000108(n), A001003(n), A007564(n), A059231(n), A078009(n), A078018(n), A081178(n), A082147(n), A082181(n), A082148(n), A082173(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 respectively .

Sum_{k, 0<=k<=n}T(n,k)*x^(n-k) = A000007(n), A000108(n), A006318(n), A047891(n+1), A082298(n), A082301(n), A082302(n), A082305(n), A082366(n), A082367(n), for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively . - Philippe Deléham, Oct 23 2007

Sum_{k, 0<=k<=[n/2]}T(n-k,k) = A004148(n). - Philippe Deléham, Nov 06 2007

T(2n,n) = A125558(n). - Philippe Deléham, Nov 16 2011

EXAMPLE

Triangle begins:

1;

1, 0;

1, 1, 0;

1, 3, 1, 0;

1, 6, 6, 1, 0;

1, 10, 20, 10, 1, 0;

1, 15, 50, 50, 15, 1, 0;

1, 21, 105, 175, 105, 21, 1, 0;

1, 28, 196, 490, 490, 196, 28, 1, 0 ;...

MAPLE

T := (n, k) -> `if`(n=0, 0^n, binomial(n, k)^2*(n-k)/(n*(k+1)));

seq(print(seq(T(n, k), k=0..n)), n=0..5); # Peter Luschny, Jun 08 2014

CROSSREFS

Cf. A000217, A002415, A006542, A006857.

Sequence in context: A165253 A059045 A122935 * A090181 A256551 A144417

Adjacent sequences:  A131195 A131196 A131197 * A131199 A131200 A131201

KEYWORD

nonn,tabl

AUTHOR

Philippe Deléham, Oct 20 2007

STATUS

approved

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Last modified October 17 22:56 EDT 2017. Contains 293484 sequences.