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URL: https://oeis.org/A115952

⇱ A115952 - OEIS

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A115952
Expansion of (1-x+x*y)/(1-x^2*y^2) - x^2/(1-x^2*y).
4
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.
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
(SageMath)
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 A380536 A370122
KEYWORD
easy,sign,tabl
AUTHOR
Paul Barry, Feb 02 2006
STATUS
approved