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A115717
A divide-and-conquer triangle related to A007583.
4
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.
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
KEYWORD
easy,sign,tabl
AUTHOR
Paul Barry, Jan 29 2006
STATUS
approved