A115717 A divide-and-conquer triangle related to A007583.

1, 0, 1, 3, -1, 1, 0, 0, 0, 1, 0, 4, -1, -1, 1, 0, 0, 0, 0, 0, 1, 12, -4, 4, 0, -1, -1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 4, 0, 0, -1, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 16, -4, -4, 4, 0, 0, 0, -1, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, -1, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 48, -16, 16, 0, -4, -4, 4, 0, 0, 0, 0, 0, -1, -1, 1

Offset

0,4

Comments

Product of (1-x, x), which is A167374, and number triangle A115715.

Links

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

Formula

Sum_{k=0..n} T(n, k) = A115716(n).

T(n ,k) = Sum_{j=k..n} A167374(n, j)*A115715(j, k). - R. J. Mathar, Sep 07 2016

Example

Triangle begins
   1;
   0,   1;
   3,  -1,  1;
   0,   0,  0,  1;
   0,   4, -1, -1,  1;
   0,   0,  0,  0,  0,  1;
  12,  -4,  4,  0, -1, -1,  1;
   0,   0,  0,  0,  0,  0,  0,  1;
   0,   0,  0,  4,  0,  0, -1, -1,  1;
   0,   0,  0,  0,  0,  0,  0,  0,  0,  1;
   0,  16, -4, -4,  4,  0,  0,  0, -1, -1,  1;
   0,   0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  1;
   0,   0,  0,  0,  0,  4,  0,  0,  0,  0, -1, -1,  1;
   0,   0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  1;
  48, -16, 16,  0, -4, -4,  4,  0,  0,  0,  0,  0, -1, -1,  1;

Maple

A115717 := proc(n, k)
    add( A167374(n, j)*A115715(j, k), j=k..n) ;
end proc: # R. J. Mathar, Sep 07 2016

Mathematica

A167374[n_, k_]:= If[k>n-2, (-1)^(n-k), 0];
g[n_, k_]:= g[n, k]= If[k==n, 1, If[k==n-1, -Mod[n, 2], If[n==2*k+2, -4, 0]]]; (* g = A115713 *)
f[n_, k_]:= f[n, k]= If[k==n, 1, -Sum[f[n, j]*g[j, k], {j, k+1, n}]]; (* f=A115715 *)
A115717[n_, k_]:= A115717[n, k]= Sum[A167374[n, j]*f[j, k], {j, k, n}];
Table[A115717[n, k], {n, 0, 15}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 23 2021 *)

Prog

(Sage)
@cached_function
def A115717(n, k):
    def A167374(n, k):
        if (k>n-2): return (-1)^(n-k)
        else: return 0
    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+k+1)*A115713(j+k+1, k) for j in (0..n-k-1) )
    return sum( A167374(n, j+k)*A115715(j+k, k) for j in (0..n-k) )
flatten([[A115717(n, k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Nov 23 2021

Crossrefs

Cf. A007583, A115715, A115716 (row sums), A167374.

Sequence in context: A278105 A074063 A338211 * A339632 A350879 A115718

Adjacent sequences: A115714 A115715 A115716 * A115718 A115719 A115720

Keyword

easy,sign,tabl

Author

Paul Barry, Jan 29 2006

Status

approved