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

0,4

Links

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

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).

Example

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;

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 *)

Prog

(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
(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

Crossrefs

Cf. A115634 (row sums), A115635 (diagonal sums), A115636 (inverse).

Sequence in context: A128131 A321188 A322076 * A115713 A199571 A036859

Adjacent sequences: A115630 A115631 A115632 * A115634 A115635 A115636

Keyword

easy,sign,tabl

Author

Paul Barry, Jan 27 2006

Status

approved