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

0,4

Links

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

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

Example

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;

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 *)

Prog

(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

Crossrefs

Cf. A032925 (row sums), A115639 (first column), A115713 (inverse).

Sequence in context: A221275 A176803 A115636 * A292143 A350824 A281441

Adjacent sequences: A115712 A115713 A115714 * A115716 A115717 A115718

Keyword

easy,nonn,tabl

Author

Paul Barry, Jan 29 2006

Status

approved