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
Examples
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; ...
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.
Programs
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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
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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
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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 *) -
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
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
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