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Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,1,1,1,1,1,1,...] DELTA [0,1,0,1,0,1,0,1,0,...] where DELTA is the operator defined in A084938.
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%I #23 Sep 08 2022 08:45:31

%S 1,1,0,2,1,0,5,5,1,0,14,21,9,1,0,42,84,56,14,1,0,132,330,300,120,20,1,

%T 0,429,1287,1485,825,225,27,1,0,1430,5005,7007,5005,1925,385,35,1,0,

%U 4862,19448,32032,28028,14014,4004,616,44,1,0,16796,75582,143208,148512,91728,34398,7644,936,54,1,0

%N Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,1,1,1,1,1,1,...] DELTA [0,1,0,1,0,1,0,1,0,...] where DELTA is the operator defined in A084938.

%C Mirror image of triangle A086810; another version of A126216.

%C Equals A131198*A007318 as infinite lower triangular matrices. - _Philippe Deléham_, Oct 23 2007

%C Diagonal sums: A119370. - _Philippe Deléham_, Nov 09 2009

%H G. C. Greubel, <a href="/A133336/b133336.txt">Table of n, a(n) for the first 50 rows, flattened</a>

%H W. Y. C. Chen, T. Mansour and S. H. F. Yan, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v13i1r112">Matchings avoiding partial patterns</a>, The Electronic Journal of Combinatorics 13, 2006, #R112, Theorem 3.3.

%F Sum_{k=0..n} T(n,k)*x^k = 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 respectively.

%F Sum_{k=0..n} T(n,k)*x^(n-k) = A000007(n), A001003(n), A107841(n), A131763(n), A131765(n), A131846(n), A131926(n), A131869(n), A131927(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8 respectively. - _Philippe Deléham_, Nov 05 2007

%F Sum_{k=0..n} T(n,k)*(-2)^k*5^(n-k) = A152601(n). - _Philippe Deléham_, Dec 10 2008

%F T(n,k) = binomial(n-1,k)*binomial(2n-k,n)/(n+1), k <= n. - _Philippe Deléham_, Nov 02 2009

%e Triangle begins:

%e 1;

%e 1, 0;

%e 2, 1, 0;

%e 5, 5, 1, 0;

%e 14, 21, 9, 1, 0;

%e 42, 84, 56, 14, 1, 0;

%e 132, 330, 300, 120, 20, 1, 0;

%e 429, 1287, 1485, 825, 225, 27, 1, 0;

%t Table[Binomial[n-1,k]*Binomial[2*n-k,n]/(n+1), {n,0,10}, {k,0,n}] // Flatten (* _G. C. Greubel_, feb 05 2018 *)

%o (PARI) for(n=0,10, for(k=0,n, print1(binomial(n-1,k)*binomial(2*n-k,n)/(n+1), ", "))) \\ _G. C. Greubel_, Feb 05 2018

%o (Magma) [[Binomial(n-1,k)*Binomial(2*n-k,n)/(n+1): k in [0..n]]: n in [0..10]]; // _G. C. Greubel_, Feb 05 2018

%Y Cf. A000108, A002054, A002055, A002056, A007160, A033280, A033281, A033282.

%K nonn,tabl

%O 0,4

%A _Philippe Deléham_, Oct 19 2007