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A131198
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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.
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7
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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
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OFFSET
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0,8
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COMMENTS
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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
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LINKS
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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) 09.7.6
Paul Barry, On a Generalization of the Narayana Triangle, J. Int. Seq. 14 (2011) # 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) # 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.
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FORMULA
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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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EXAMPLE
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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; ...
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MAPLE
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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
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A000217, A002415, A006542, A006857.
Sequence in context: A059045 A348210 A122935 * A090181 A256551 A144417
Adjacent sequences: A131195 A131196 A131197 * A131199 A131200 A131201
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KEYWORD
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nonn,tabl
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AUTHOR
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Philippe Deléham, Oct 20 2007
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STATUS
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approved
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