A115633 A divide and conquer-related triangle: see formula for T(n,k), n >= k >= 0.
1, 1, -1, -4, 0, 1, 0, 0, 1, -1, 0, -4, 0, 0, 1, 0, 0, 0, 0, 1, -1, 0, 0, -4, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, -4, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0, 1
Offset: 0
Examples
Triangle begins 1; 1, -1; -4, 0, 1; 0, 0, 1, -1; 0, -4, 0, 0, 1; 0, 0, 0, 0, 1, -1; 0, 0, -4, 0, 0, 0, 1; 0, 0, 0, 0, 0, 0, 1, -1; 0, 0, 0, -4, 0, 0, 0, 0, 1; 0, 0, 0, 0, 0, 0, 0, 0, 1, -1; 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, 1; 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1; 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0, 1; 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1; 0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0, 1; 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1; 0, 0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0, 0, 1;
Links
- G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Programs
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Mathematica
T[n_, k_]:= If[k==n, (-1)^n, If[k==n-1, (1-(-1)^n)/2, If[n==2*k+2, -4, 0]]]; Table[T[n, k], {n,0,18}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 23 2021 *) -
PARI
A115633(n,k)=if(n==k, (-1)^n, bittest(n,0), k==n-1, k+1==n\2, -4) \\ M. F. Hasler, Nov 24 2021
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Sage
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 flatten([[A115633(n,k) for k in (0..n)] for n in (0..18)]) # G. C. Greubel, Nov 23 2021
Formula
T(n, k) = (-1)^n if n = k; else -4 if n = 2k+2; else (n mod 2) if n = k+1; else 0.
G.f.: (1+x-x*y)/(1-x^2*y^2) - 4*x^2/(1-x^2*y).
(1, -x) + (x, x)/2 + (x, -x)/2 - 4(x^2, x^2) expressed in the notation of stretched Riordan arrays.
Column k has g.f.: (-x)^k + (x*(-x)^k + x^(k+1))/2 - 4*x^(2*k+2).
Sum_{k=0..n} T(n, k) = A115634(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A115635(n).