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URL: https://oeis.org/A115636

⇱ A115636 - OEIS

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A115636
A divide-and-conquer number triangle.
4
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
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
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;
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 *)
PROG
(SageMath)
@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
CROSSREFS
Cf. A115633 (inverse), A115637 (row sums), A115639 (first column).
Sequence in context: A122873 A221275 A176803 * A115715 A292143 A350824
KEYWORD
sign,tabl
AUTHOR
Paul Barry, Jan 27 2006
STATUS
approved