A115715 A divide-and-conquer 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 29 2006

Keywords

• easy
• nonn
• 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. A032925 (row sums), A115639 (first column), A115713 (inverse).

Programs

Maple

A115715 := proc(n,k)
    option remember;
    if n = k then
        1;
    elif k > n then
        0;
    else
        -add(procname(n,l)*A115713(l,k),l=k+1..n) ;
    end if;
end proc:
seq(seq(A115715(n,k),k=0..n),n=0..13) ; # R. J. Mathar, Sep 07 2016

Mathematica

A115713[n_, k_]:= If[k==n, 1, If[k==n-1, ((-1)^n-1)/2, If[n==2*k+2, -4, 0]]];
T[n_, k_]:= T[n, k]= If[k==n, 1, -Sum[T[n, j]*A115713[j, k], {j, k+1, n}]];
Table[T[n, k], {n,0,18}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 23 2021 *)

Sage

@CachedFunction
def A115713(n,k):
    if (k==n): return 1
    elif (k==n-1): return -(n%2)
    elif (n==2*k+2): return -4
    else: return 0
def A115715(n,k):
    if (k==0): return 4^(floor(log(n+2, 2)) -1)
    elif (k==n): return 1
    elif (k==n-1): return (n%2)
    else: return (-1)*sum( A115715(n,j)*A115713(j,k) for j in (k+1..n) )
flatten([[A115715(n,k) for k in (0..n)] for n in (0..18)]) # G. C. Greubel, Nov 23 2021

Formula

Sum_{=0..n} T(n, k) = A032925(n).

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

T(n, k) = 1 if n = k, otherwise T(n, k) = (-1)*Sum_{j=k+1..n} T(n, j)*A115713(j, k). - R. J. Mathar, Sep 07 2016