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

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.

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

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

Cf. A115713, A115716, A115717, A237420.

Sequence in context: A115717 A339632 A350879 * A361510 A204181 A204242

Adjacent sequences: A115715 A115716 A115717 * A115719 A115720 A115721

Keyword

easy,sign,tabl

Author

Paul Barry, Jan 29 2006

Status

approved