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A115718
Inverse of number triangle A115717; a divide-and-conquer related triangle.
1
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, 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, 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
OFFSET
0,4
COMMENTS
Product of A115713 and (1/(1-x), x).
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.
Row sums of inverse are A115716.
FORMULA
From G. C. Greubel, Nov 29 2021: (Start)
T(2*n, k) = -3 if (k < n/2) otherwise 1.
T(2*n+1, k) = 0 if (k < n) otherwise 1.
Sum_{k=0..n} T(n, k) = (1/2)*(2 + (1 + (-1)^n)*n) = 1 + A237420(n). (End)
EXAMPLE
Triangle begins
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, 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, 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;
-3, -3, -3, -3, -3, -3, -3, 1, 1, 1, 1, 1, 1, 1, 1;
MATHEMATICA
T[n_, k_]:= If[OddQ[n], If[k<n, 0, 1], If[k<n/2, -3, 1]];
Table[T[n, k], {n, 0, 16}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 29 2021 *)
PROG
(Sage)
def A115718(n, k):
if (n%2==0): return 0 if (k<n) else 1
else: return -3 if (k<n/2) else 1
flatten([[A115718(n, k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Nov 29 2021
CROSSREFS
KEYWORD
easy,sign,tabl
AUTHOR
Paul Barry, Jan 29 2006
STATUS
approved