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A115713 A divide-and-conquer related triangle.

%I A115713 #8 Nov 24 2021 03:06:44
%S A115713 1,-1,1,-4,0,1,0,0,-1,1,0,-4,0,0,1,0,0,0,0,-1,1,0,0,-4,0,0,0,1,0,0,0,
%T A115713 0,0,0,-1,1,0,0,0,-4,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,-4,0,0,0,
%U A115713 0,0,1,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,-4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-4,0,0,0,0,0,0,0,1
%N A115713 A divide-and-conquer related triangle.
%H A115713 G. C. Greubel, <a href="/A115713/b115713.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A115713 G.f.: (1-x+x*y)/(1-x^2*y^2) - 4*x^2/(1-x^2*y).
%F A115713 (1, x) - (x, x)/2 - (x, -x)/2 - 4*(x^2, x^2) expressed in the notation of stretched Riordan arrays.
%F A115713 Column k has g.f.: x^k - (x*(-x)^k + x^(k+1))/2 - 4*x^(2*k+2).
%F A115713 T(n, k) = if(n=k, 1, 0) OR if(n=2k+2, -4, 0) OR if(n=k+1, -(1+(-1)^k)/2, 0).
%F A115713 Sum_{k=0..n} T(n, k) = A115634(n).
%F A115713 Sum_{k=0..floor(n/2)} T(n-k, k) = A115714(n).
%e A115713 Triangle begins
%e A115713    1;
%e A115713   -1,  1;
%e A115713   -4,  0,  1;
%e A115713    0,  0, -1,  1;
%e A115713    0, -4,  0,  0,  1;
%e A115713    0,  0,  0,  0, -1,  1;
%e A115713    0,  0, -4,  0,  0,  0,  1;
%e A115713    0,  0,  0,  0,  0,  0, -1,  1;
%e A115713    0,  0,  0, -4,  0,  0,  0,  0,  1;
%e A115713    0,  0,  0,  0,  0,  0,  0,  0, -1,  1;
%e A115713    0,  0,  0,  0, -4,  0,  0,  0,  0,  0,  1;
%e A115713    0,  0,  0,  0,  0,  0,  0,  0,  0,  0, -1,  1;
%e A115713    0,  0,  0,  0,  0, -4,  0,  0,  0,  0,  0,  0,  1;
%e A115713    0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0, -1,  1;
%e A115713    0,  0,  0,  0,  0,  0, -4,  0,  0,  0,  0,  0,  0,  0,  1;
%p A115713 A115713 := proc(n,k)
%p A115713     coeftayl( (1-x+x*y)/(1-x^2*y^2)-4*x^2/(1-x^2*y),x=0,n) ;
%p A115713     coeftayl( %,y=0,k) ;
%p A115713 end proc: # _R. J. Mathar_, Sep 07 2016
%t A115713 T[n_, k_]:= If[k==n, 1, If[k==n-1, -(1-(-1)^n)/2, If[n==2*k+2, -4, 0]]];
%t A115713 Table[T[n, k], {n,0,18}, {k,0,n}]//Flatten (* _G. C. Greubel_, Nov 23 2021 *)
%o A115713 (Sage)
%o A115713 def A115713(n,k):
%o A115713     if (k==n): return 1
%o A115713     elif (k==n-1): return -(n%2)
%o A115713     elif (n==2*k+2): return -4
%o A115713     else: return 0
%o A115713 flatten([[A115713(n,k) for k in (0..n)] for n in (0..18)]) # _G. C. Greubel_, Nov 23 2021
%Y A115713 Cf. A115634 (row sums), A115714 (diagonal sums), A115715 (inverse).
%K A115713 easy,sign,tabl
%O A115713 0,4
%A A115713 _Paul Barry_, Jan 29 2006