A115636 A divide-and-conquer number triangle.

1, 1, -1, 4, 0, 1, 4, 0, 1, -1, 4, -4, 0, 0, 1, 4, -4, 0, 0, 1, -1, 16, 0, 4, 0, 0, 0, 1, 16, 0, 4, 0, 0, 0, 1, -1, 16, 0, 4, -4, 0, 0, 0, 0, 1, 16, 0, 4, -4, 0, 0, 0, 0, 1, -1, 16, -16, 0, 0, 4, 0, 0, 0, 0, 0, 1, 16, -16, 0, 0, 4, 0, 0, 0, 0, 0, 1, -1, 16, -16, 0, 0, 4, -4, 0, 0, 0, 0, 0, 0, 1, 16, -16, 0, 0, 4, -4, 0, 0, 0, 0, 0, 0, 1, -1

Offset: 0

Author

Paul Barry, Jan 27 2006

Keywords

• sign
• tabl

Examples

			Triangle begins
   1;
   1,  -1;
   4,   0,  1;
   4,   0,  1, -1;
   4,  -4,  0,  0,  1;
   4,  -4,  0,  0,  1, -1;
  16,   0,  4,  0,  0,  0,  1;
  16,   0,  4,  0,  0,  0,  1, -1;
  16,   0,  4, -4,  0,  0,  0,  0,  1;
  16,   0,  4, -4,  0,  0,  0,  0,  1, -1;
  16, -16,  0,  0,  4,  0,  0,  0,  0,  0,  1;
  16, -16,  0,  0,  4,  0,  0,  0,  0,  0,  1, -1;
  16, -16,  0,  0,  4, -4,  0,  0,  0,  0,  0,  0,  1;
  16, -16,  0,  0,  4, -4,  0,  0,  0,  0,  0,  0,  1, -1;
  64,   0, 16,  0,  0,  0,  4,  0,  0,  0,  0,  0,  0,  0,  1;
  64,   0, 16,  0,  0,  0,  4,  0,  0,  0,  0,  0,  0,  0,  1, -1;
  64,   0, 16,  0,  0,  0,  4, -4,  0,  0,  0,  0,  0,  0,  0,  0,  1;
		

Links

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

Crossrefs

Cf. A115633 (inverse), A115637 (row sums), A115639 (first column).

Programs

Mathematica

A115633[n_, k_]:= If[k==n, (-1)^n, If[k==n-1, Mod[n,2], If[n==2*k+2, -4, 0]]];
T[n_, k_]:= T[n, k]= (-1)^k*If[k==n, 1, -Sum[T[n, j]*A115633[j, k], {j,k+1,n}] ];
Table[T[n, k], {n,0,18}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 24 2021 *)

Sage

@CachedFunction
def A115633(n, k):
    if (k==n): return (-1)^n
    elif (k==n-1): return n%2
    elif (n==2*k+2): return -4
    else: return 0
def A115636(n,k):
    if (k==0): return 4^(floor(log(n+2, 2)) -1)
    elif (k==n): return (-1)^n
    elif (k==n-1): return (n%2)
    else: return (-1)^(k+1)*sum( A115636(n, j)*A115633(j, k) for j in (k+1..n) )
flatten([[A115636(n, k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Nov 24 2021

Formula

T(n, 0) = A115639(n).

Sum_{k=0..n} T(n, k) = A115637(n).

T(n, k) = (-1)^k*( 1 if k = n otherwise (-1)*Sum_{j=k+1..n} T(n, j)*A115633(j, k) ). - G. C. Greubel, Nov 24 2021