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%I #49 Jul 14 2023 04:18:07
%S 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,
%T 105,175,105,21,1,0,1,28,196,490,490,196,28,1,0,1,36,336,1176,1764,
%U 1176,336,36,1,0,1,45,540,2520,5292,5292,2520,540,45,1,0
%N 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.
%C Mirror image of triangle A090181, another version of triangle of Narayana (A001263).
%C Equals A133336*A130595 as infinite lower triangular matrices. - _Philippe Deléham_, Oct 23 2007
%H G. C. Greubel, <a href="/A131198/b131198.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Barry3/barry93.html">Continued fractions and transformations of integer sequences</a>, JIS 12 (2009), Article 09.7.6.
%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Barry4/barry142.html">On a Generalization of the Narayana Triangle</a>, J. Int. Seq. 14 (2011), Article 11.4.5.
%H Paul Barry, <a href="https://arxiv.org/abs/1802.03443">On a transformation of Riordan moment sequences</a>, arXiv:1802.03443 [math.CO], 2018.
%H Paul Barry and A. Hennessy, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Barry2/barry126.html">A Note on Narayana Triangles and Related Polynomials, Riordan Arrays, and MIMO Capacity Calculations</a>, J. Int. Seq. 14 (2011), Article 11.3.8.
%H FindStat - Combinatorial Statistic Finder, <a href="http://www.findstat.org/StatisticsDatabase/St000015">The number of peaks of a Dyck path.</a>, <a href="http://www.findstat.org/StatisticsDatabase/St000024">The number of double rises of a Dyck path.</a>, <a href="http://www.findstat.org/StatisticsDatabase/St000053">The number of valleys of a Dyck path.</a>, <a href="http://www.findstat.org/StatisticsDatabase/St000083">The number of left oriented leafs except the first one of a binary tree.</a>, <a href="http://www.findstat.org/StatisticsDatabase/St000120">The number of left tunnels of a Dyck path.</a>
%H Aoife Hennessy, <a href="http://repository.wit.ie/1693/1/AoifeThesis.pdf">A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths</a>, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
%F 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.
%F 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
%F Sum_{k=0..floor(n/2)} T(n-k,k) = A004148(n). - _Philippe Deléham_, Nov 06 2007
%F T(2*n,n) = A125558(n). - _Philippe Deléham_, Nov 16 2011
%F T(n, k) = [x^k] hypergeom([1 - n, -n], [2], x). - _Peter Luschny_, Apr 26 2022
%e Triangle begins:
%e 1;
%e 1, 0;
%e 1, 1, 0;
%e 1, 3, 1, 0;
%e 1, 6, 6, 1, 0;
%e 1, 10, 20, 10, 1, 0;
%e 1, 15, 50, 50, 15, 1, 0;
%e 1, 21, 105, 175, 105, 21, 1, 0;
%e 1, 28, 196, 490, 490, 196, 28, 1, 0; ...
%p T := (n,k) -> `if`(n=0, 0^n, binomial(n,k)^2*(n-k)/(n*(k+1)));
%p seq(print(seq(T(n,k), k=0..n)), n=0..5); # _Peter Luschny_, Jun 08 2014
%p R := n -> simplify(hypergeom([1 - n, -n], [2], x)):
%p Trow := n -> seq(coeff(R(n, x), x, k), k = 0..n):
%p seq(print(Trow(n)), n = 0..9); # _Peter Luschny_, Apr 26 2022
%t 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 *)
%o (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
%o (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
%Y Cf. A000217, A002415, A006542, A006857.
%K nonn,tabl
%O 0,8
%A _Philippe Deléham_, Oct 20 2007
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