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A115718 Inverse of number triangle A115717; a divide-and-conquer related triangle.

%I A115718 #8 Nov 30 2021 05:36:10
%S A115718 1,0,1,-3,1,1,0,0,0,1,-3,-3,1,1,1,0,0,0,0,0,1,-3,-3,-3,1,1,1,1,0,0,0,
%T A115718 0,0,0,0,1,-3,-3,-3,-3,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,-3,-3,-3,-3,-3,1,
%U A115718 1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,-3,-3,-3,-3,-3,-3,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1
%N A115718 Inverse of number triangle A115717; a divide-and-conquer related triangle.
%C A115718 Product of A115713 and (1/(1-x), x).
%C A115718 Row sums are 1,1,-1,1,-3,1,-5,1,-7,1, ... with g.f. (1+x-3*x^2-x^3)/(1-x^2)^2.
%C A115718 Row sums of inverse are A115716.
%H A115718 G. C. Greubel, <a href="/A115718/b115718.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A115718 From _G. C. Greubel_, Nov 29 2021: (Start)
%F A115718 T(2*n, k) = -3 if (k < n/2) otherwise 1.
%F A115718 T(2*n+1, k) = 0 if (k < n) otherwise 1.
%F A115718 Sum_{k=0..n} T(n, k) = (1/2)*(2 + (1 + (-1)^n)*n) = 1 + A237420(n). (End)
%e A115718 Triangle begins
%e A115718    1;
%e A115718    0,  1;
%e A115718   -3,  1,  1;
%e A115718    0,  0,  0,  1;
%e A115718   -3, -3,  1,  1,  1;
%e A115718    0,  0,  0,  0,  0,  1;
%e A115718   -3, -3, -3,  1,  1,  1,  1;
%e A115718    0,  0,  0,  0,  0,  0,  0,  1;
%e A115718   -3, -3, -3, -3,  1,  1,  1,  1,  1;
%e A115718    0,  0,  0,  0,  0,  0,  0,  0,  0,  1;
%e A115718   -3, -3, -3, -3, -3,  1,  1,  1,  1,  1,  1;
%e A115718    0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  1;
%e A115718   -3, -3, -3, -3, -3, -3,  1,  1,  1,  1,  1,  1,  1;
%e A115718    0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  1;
%e A115718   -3, -3, -3, -3, -3, -3, -3,  1,  1,  1,  1,  1,  1,  1,  1;
%t A115718 T[n_, k_]:= If[OddQ[n], If[k<n, 0, 1], If[k<n/2, -3, 1]];
%t A115718 Table[T[n, k], {n,0,16}, {k,0,n}]//Flatten (* _G. C. Greubel_, Nov 29 2021 *)
%o A115718 (Sage)
%o A115718 def A115718(n,k):
%o A115718     if (n%2==0): return 0 if (k<n) else 1
%o A115718     else: return -3 if (k<n/2) else 1
%o A115718 flatten([[A115718(n,k) for k in (0..n)] for n in (0..15)]) # _G. C. Greubel_, Nov 29 2021
%Y A115718 Cf. A115713, A115716, A115717, A237420.
%K A115718 easy,sign,tabl
%O A115718 0,4
%A A115718 _Paul Barry_, Jan 29 2006