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.

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

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

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

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

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

Paul Barry, On a transformation of Riordan moment sequences, arXiv:1802.03443 [math.CO], 2018.

Paul Barry and A. Hennessy, A Note on Narayana Triangles and Related Polynomials, Riordan Arrays, and MIMO Capacity Calculations, J. Int. Seq. 14 (2011), Article 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..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..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..floor(n/2)} T(n-k,k) = A004148(n). - Philippe Deléham, Nov 06 2007

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

T(n, k) = [x^k] hypergeom([1 - n, -n], [2], x). - Peter Luschny, Apr 26 2022

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
R := n -> simplify(hypergeom([1 - n, -n], [2], x)):
Trow := n -> seq(coeff(R(n, x), x, k), k = 0..n):
seq(print(Trow(n)), n = 0..9); # Peter Luschny, Apr 26 2022

Mathematica

Table[If[n == 0, 1, (n-k)*Binomial[n, k]^2/(n*(k+1))], {n, 0, 10}, {k, 0, n}] //Flatten (* G. C. Greubel, Feb 06 2018 *)

Prog

(PARI) for(n=0, 10, for(k=0, n, print1(if(n==0, 1, (n-k)*binomial(n, k)^2/(n* (k+1))), ", "))) \\ G. C. Greubel, Feb 06 2018
(Magma) [[n le 0 select 1 else (n-k)*Binomial(n, k)^2/(n*(k+1)): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Feb 06 2018

Crossrefs

Cf. A000217, A002415, A006542, A006857.

Sequence in context: A059045 A348210 A122935 * A090181 A368504 A256551

Adjacent sequences: A131195 A131196 A131197 * A131199 A131200 A131201

Keyword

nonn,tabl

Author

Philippe Deléham, Oct 20 2007

Status

approved