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%I #16 Sep 08 2022 08:45:54
%S 1,1,-1,1,0,-1,1,2,-2,-1,1,5,0,-5,-1,1,9,10,-10,-9,-1,1,14,35,0,-35,
%T -14,-1,1,20,84,70,-70,-84,-20,-1,1,27,168,294,0,-294,-168,-27,-1,1,
%U 35,300,840,588,-588,-840,-300,-35,-1
%N Triangle which contains the first differences of the Catalan triangle A001263 constructed along rows.
%H G. C. Greubel, <a href="/A178655/b178655.txt">Rows n = 0..100 of triangle, flattened</a>
%F T(n,k) = -T(n,n-k), n > 0.
%F T(n,k) = A001263(n,k+1) - A001263(n,k), n > 0. - _R. J. Mathar_, Jun 16 2015
%e Triangle begins
%e 1;
%e 1, -1;
%e 1, 0, -1;
%e 1, 2, -2, -1;
%e 1, 5, 0, -5, -1;
%e 1, 9, 10, -10, -9, -1;
%e 1, 14, 35, 0, -35, -14, -1;
%e 1, 20, 84, 70, -70, -84, -20, -1;
%e 1, 27, 168, 294, 0, -294, -168, -27, -1;
%e 1, 35, 300, 840, 588, -588, -840, -300, -35, -1;
%t Join[{1}, Table[((n+1)*(n-2*k)/(n*(k+1)*(n-k+1)))*Binomial[n, k]^2, {n, 1, 10}, {k, 0, n}]//Flatten] (* _G. C. Greubel_, Jan 28 2019 *)
%o (PARI) {T(n,k) = if(n==0, 1, ((n+1)*(n-2*k)/(n*(k+1)*(n-k+1)))* binomial(n, k)^2)};
%o for(n=0,10, for(k=0,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, Jan 28 2019
%o (Magma) [[n le 0 select 1 else ((n+1)*(n-2*k)/(n*(k+1)*(n-k+1)))*Binomial(n, k)^2: k in [0..n]]: n in [0..10]]; // _G. C. Greubel_, Jan 28 2019
%o (Sage) [1] + [[((n+1)*(n-2*k)/(n*(k+1)*(n-k+1)))* binomial(n, k)^2 for k in (0..n)] for n in (1..10)] # _G. C. Greubel_, Jan 28 2019
%Y Cf. A001263, A000007 (row sums).
%K sign,tabl,easy
%O 0,8
%A _Roger L. Bagula_, Jun 01 2010
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