A115634 -id:A115634 - cp's OEIS frontend

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

Paul Barry, Jan 27 2006

			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;

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

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

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 *)

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

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

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).

A115713 A divide-and-conquer related triangle.

Original entry on oeis.org

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

Paul Barry, Jan 29 2006

			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;

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

Cf. A115634 (row sums), A115714 (diagonal sums), A115715 (inverse).

A115713 := proc(n,k)
    coeftayl( (1-x+x*y)/(1-x^2*y^2)-4*x^2/(1-x^2*y),x=0,n) ;
    coeftayl( %,y=0,k) ;
end proc: # R. J. Mathar, Sep 07 2016

T[n_, k_]:= If[k==n, 1, 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 *)

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
flatten([[A115713(n,k) for k in (0..n)] for n in (0..18)]) # G. C. Greubel, Nov 23 2021

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).
T(n, k) = if(n=k, 1, 0) OR if(n=2k+2, -4, 0) OR if(n=k+1, -(1+(-1)^k)/2, 0).
Sum_{k=0..n} T(n, k) = A115634(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A115714(n).

cp's OEIS Frontend

A115633 A divide and conquer-related triangle: see formula for T(n,k), n >= k >= 0.

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