A115952 Expansion of (1-x+x*y)/(1-x^2*y^2) - x^2/(1-x^2*y).

1, -1, 1, -1, 0, 1, 0, 0, -1, 1, 0, -1, 0, 0, 1, 0, 0, 0, 0, -1, 1, 0, 0, -1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, -1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, -1, 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, -1

Offset

0,1

Comments

Row sums are A000007. Diagonal sums are A115953. Inverse is A115954.

Links

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Formula

Number triangle T(n,k)=if(n=k,1,0) OR if(n=2k+2,-1,0) OR if(n=k+1,-(1+(-1)^k)/2,0).

Example

Triangle begins
   1,
  -1,  1,
  -1,  0,  1,
   0,  0, -1,  1,
   0, -1,  0,  0,  1,
   0,  0,  0,  0, -1,  1,
   0,  0, -1,  0,  0,  0,  1,
   0,  0,  0,  0,  0,  0, -1,  1,
   0,  0,  0, -1,  0,  0,  0,  0,  1,
   0,  0,  0,  0,  0,  0,  0,  0, -1,  1,
   0,  0,  0,  0, -1,  0,  0,  0,  0,  0,  1,
   0,  0,  0,  0,  0,  0,  0,  0,  0,  0, -1,  1,
   0,  0,  0,  0,  0, -1,  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, -1,  0,  0,  0,  0,  0,  0,  0,  1,
   0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0, -1,  1

Mathematica

T[n_, k_]:= If[n==k, 1, If[n==k+1, -(1+(-1)^k)/2, If[n==2*k+2, -1, 0]]];
Table[T[n, k], {n, 0, 15}, {k, 0, n}]//Flatten (* G. C. Greubel, May 06 2019 *)

Prog

(PARI) {T(n, k) = if(n==k, 1, if(n==k+1, -(1+(-1)^k)/2, if(n==2*k+2, -1, 0)))}; \\ G. C. Greubel, May 06 2019
(Magma) [[n eq k select 1 else n eq k+1 select -(1+(-1)^k)/2 else n eq 2*(k+1) select -1 else 0: k in [0..n]]: n in [0..15]]; // G. C. Greubel, May 06 2019
(Sage)
def T(n, k):
    if (n==k): return 1
    elif (n==k+1): return -(1+(-1)^k)/2
    elif (n==2*(k+1)): return -1
    else: return 0
[[T(n, k) for k in (0..n)] for n in (0..15)] # G. C. Greubel, May 06 2019

Crossrefs

Cf. A115524.

Sequence in context: A283437 A155898 A181650 * A115524 A370122 A117198

Adjacent sequences: A115949 A115950 A115951 * A115953 A115954 A115955

Keyword

easy,sign,tabl

Author

Paul Barry, Feb 02 2006

Status

approved