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A133336 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.

%I A133336 #28 Jul 13 2025 11:02:05
%S A133336 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 A133336 0,429,1287,1485,825,225,27,1,0,1430,5005,7007,5005,1925,385,35,1,0,
%U A133336 4862,19448,32032,28028,14014,4004,616,44,1,0,16796,75582,143208,148512,91728,34398,7644,936,54,1,0
%N A133336 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 A133336 Mirror image of triangle A086810; another version of A126216.
%C A133336 Equals A131198*A007318 as infinite lower triangular matrices. - _Philippe Deléham_, Oct 23 2007
%C A133336 Diagonal sums: A119370. - _Philippe Deléham_, Nov 09 2009
%H A133336 G. C. Greubel, <a href="/A133336/b133336.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H A133336 W. Y. C. Chen, T. Mansour and S. H. F. Yan, <a href="https://doi.org/10.37236/1138">Matchings avoiding partial patterns</a>, The Electronic Journal of Combinatorics 13, 2006, #R112, Theorem 3.3.
%F A133336 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 A133336 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 A133336 Sum_{k=0..n} T(n,k)*(-2)^k*5^(n-k) = A152601(n). - _Philippe Deléham_, Dec 10 2008
%F A133336 T(n,k) = binomial(n-1,k)*binomial(2n-k,n)/(n+1), k <= n. - _Philippe Deléham_, Nov 02 2009
%e A133336 Triangle begins:
%e A133336     1;
%e A133336     1,    0;
%e A133336     2,    1,    0;
%e A133336     5,    5,    1,   0;
%e A133336    14,   21,    9,   1,   0;
%e A133336    42,   84,   56,  14,   1,  0;
%e A133336   132,  330,  300, 120,  20,  1, 0;
%e A133336   429, 1287, 1485, 825, 225, 27, 1, 0;
%t A133336 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 A133336 (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 A133336 (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 A133336 Cf. A000108, A002054, A002055, A002056, A007160, A033280, A033281, A033282.
%K A133336 nonn,tabl
%O A133336 0,4
%A A133336 _Philippe Deléham_, Oct 19 2007