%I #15 May 18 2021 04:22:29
%S 1,-1,1,0,-1,1,-1,1,-1,1,-2,0,2,-1,1,-6,2,1,3,-1,1,-18,5,7,2,4,-1,1,
%T -57,17,19,13,3,5,-1,1,-186,56,64,36,20,4,6,-1,1,-622,190,212,124,56,
%U 28,5,7,-1,1,-2120,654,722,416,198,79,37,6,8,-1,1,-7338,2282,2494,1434,673,287,105,47,7,9,-1,1
%N Triangle T(n,k), 0<=k<=n, read by rows given by :[ -1,1,1,1,1,1,1,...] DELTA [1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.
%C Riordan array (2/(3-sqrt(1-4*x)), (1-sqrt(1-4*x))/(3-sqrt(1-4*x))). - _Philippe Deléham_, Jan 27 2014
%H G. C. Greubel, <a href="/A127543/b127543.txt">Rows n = 0..50 of the triangle, flattened</a>
%F T(n,k) = A065600(n-1,k-1) - A065600(n-1,k).
%F Sum_{k=0..n} T(n,k)*x^k = A127053(n), A126985(n), A127016(n), A127017(n), A126987(n), A126986(n), A126982(n), A126984(n), A126983(n), A000007(n), A000108(n), A000984(n), A007854(n), A076035(n), A076036(n), A127628(n), A126694(n), A115970(n) for n= -8,-7,...,8,9 respectively.
%F Sum_{j>=0} T(n,j)*A007318(j,k) = A106566(n,k).
%F Sum_{j>=0} T(n,j)*A038207(j,k) = A039599(n,k).
%F Sum_{j>=0} T(n,j)*A027465(j,k) = A116395(n,k).
%e Triangle begins:
%e 1;
%e -1, 1;
%e 0, -1, 1;
%e -1, 1, -1, 1;
%e -2, 0, 2, -1, 1;
%e -6, 2, 1, 3, -1, 1;
%e -18, 5, 7, 2, 4, -1, 1;
%e -57, 17, 19, 13, 3, 5, -1, 1;
%t A065600[n_, k_]:= If[k==n, 1, Sum[j*Binomial[k+j, j]*Binomial[2*(n-k-j), n-k]/(n-k-j), {j,0, Floor[(n-k)/2]}]];
%t A127543[n_, k_]:= A065600[n-1,k-1] - A065600[n-1,k];
%t Table[A127543[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, May 17 2021 *)
%o (Sage)
%o def A065600(n,k): return 1 if (k==n) else sum( j*binomial(k+j, j)*binomial(2*(n-k-j), n-k)/(n-k-j) for j in (0..(n-k)//2) )
%o def A127543(n,k): return A065600(n-1, k-1) - A065600(n-1, k)
%o flatten([[A127543(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, May 17 2021
%K sign,tabl
%O 0,11
%A _Philippe Deléham_, Apr 01 2007