cp's OEIS Frontend

A127543 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.

%I A127543 #15 May 18 2021 04:22:29
%S A127543 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 A127543 -57,17,19,13,3,5,-1,1,-186,56,64,36,20,4,6,-1,1,-622,190,212,124,56,
%U A127543 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 A127543 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 A127543 Riordan array (2/(3-sqrt(1-4*x)), (1-sqrt(1-4*x))/(3-sqrt(1-4*x))). - _Philippe Deléham_, Jan 27 2014
%H A127543 G. C. Greubel, <a href="/A127543/b127543.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A127543 T(n,k) = A065600(n-1,k-1) - A065600(n-1,k).
%F A127543 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 A127543 Sum_{j>=0} T(n,j)*A007318(j,k) = A106566(n,k).
%F A127543 Sum_{j>=0} T(n,j)*A038207(j,k) = A039599(n,k).
%F A127543 Sum_{j>=0} T(n,j)*A027465(j,k) = A116395(n,k).
%e A127543 Triangle begins:
%e A127543     1;
%e A127543    -1,  1;
%e A127543     0, -1,  1;
%e A127543    -1,  1, -1,  1;
%e A127543    -2,  0,  2, -1,  1;
%e A127543    -6,  2,  1,  3, -1,  1;
%e A127543   -18,  5,  7,  2,  4, -1,  1;
%e A127543   -57, 17, 19, 13,  3,  5, -1, 1;
%t A127543 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 A127543[n_, k_]:= A065600[n-1,k-1] - A065600[n-1,k];
%t A127543 Table[A127543[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, May 17 2021 *)
%o A127543 (Sage)
%o A127543 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 A127543 def A127543(n,k): return A065600(n-1, k-1) - A065600(n-1, k)
%o A127543 flatten([[A127543(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, May 17 2021
%K A127543 sign,tabl
%O A127543 0,11
%A A127543 _Philippe Deléham_, Apr 01 2007