A115524 -id:A115524 - cp's OEIS frontend

A115526 Inverse of number triangle A115524.

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

Offset: 0

Paul Barry, Jan 25 2006

Row sums are number of zeros in binary expansion of n+2, that is, A023416(n+2) [conjecture].

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

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

Original entry on oeis.org

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

Paul Barry, Feb 02 2006

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

			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

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

Cf. A115524.

[[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

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 *)

{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

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

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).

A115525 Periodic {1,1,-2,0,1,0,-1,0,0,1,-1,-1}.

Original entry on oeis.org

1, 1, -2, 0, 1, 0, -1, 0, 0, 1, -1, -1, 1, 1, -2, 0, 1, 0, -1, 0, 0, 1, -1, -1, 1, 1, -2, 0, 1, 0, -1, 0, 0, 1, -1, -1, 1, 1, -2, 0, 1, 0, -1, 0, 0, 1, -1, -1, 1, 1, -2, 0, 1, 0, -1, 0, 0, 1, -1, -1, 1, 1, -2, 0, 1, 0, -1, 0, 0, 1, -1, -1

Offset: 0

Paul Barry, Jan 25 2006

Diagonal sums of number triangle A115524.

Index entries for linear recurrences with constant coefficients, signature (-1,-1,0,1,1,1).

PadRight[{},120,{1,1,-2,0,1,0,-1,0,0,1,-1,-1}] (* Harvey P. Dale, Aug 20 2021 *)

G.f.: (1+2x-x^3-2x^4-x^5)/(1+x+x^2-x^4-x^5-x^6).

cp's OEIS Frontend

A115526 Inverse of number triangle A115524.

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

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A115525 Periodic {1,1,-2,0,1,0,-1,0,0,1,-1,-1}.

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cp's OEIS Frontend

A115526 Inverse of number triangle A115524.

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Examples

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

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Magma

Mathematica

PARI

Sage

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A115525 Periodic {1,1,-2,0,1,0,-1,0,0,1,-1,-1}.

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Mathematica

Formula

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