A165507 -id:A165507 - cp's OEIS frontend

A165795 Array A(n, k) = numerator of 1/n^2 - 1/k^2 with A(0,k) = 1 and A(n,0) = -1, read by antidiagonals.

1, -1, 1, -1, 0, 1, -1, -3, 3, 1, -1, -8, 0, 8, 1, -1, -15, -5, 5, 15, 1, -1, -24, -3, 0, 3, 24, 1, -1, -35, -21, -7, 7, 21, 35, 1, -1, -48, -2, -16, 0, 16, 2, 48, 1, -1, -63, -45, -1, -9, 9, 1, 45, 63, 1, -1, -80, -15, -40, -5, 0, 5, 40, 15, 80, 1, -1, -99, -77, -55, -33, -11, 11, 33, 55, 77, 99, 1

Offset: 0

Paul Curtz, Sep 27 2009

A row of A(0,k)= 1 is added on top of the array shown in A172157, which is then read upwards by antidiagonals.

One may also interpret this as appending a 1 to each row of A173651 or adding a column of -1's and a diagonal of +1's to A165507.

			The array, A(n, k), of numerators starts in row n=0 with columns m>=0 as:
  .1...1...1...1...1...1...1...1...1...1...1.
  -1...0...3...8..15..24..35..48..63..80..99. A005563, A147998
  -1..-3...0...5...3..21...2..45..15..77...6. A061037, A070262
  -1..-8..-5...0...7..16...1..40..55...8..91. A061039
Antidiagonal triangle, T(n, k), begins as:
   1;
  -1,   1;
  -1,   0,   1;
  -1,  -3,   3,  1;
  -1,  -8,   0,  8,  1;
  -1, -15,  -5,  5, 15,  1;
  -1, -24,  -3,  0,  3, 24,  1;
  -1, -35, -21, -7,  7, 21, 35, 1;

G. C. Greubel, Antidiagonals n = 0..50, flattened

Cf. A158388, A165507, A172157, A173651.

Cf. A005408, A005563, A061037, A061039, A070262, A147998.

T[n_, k_]:= If[k==n, 1, If[k==0, -1, Numerator[1/(n-k)^2 - 1/k^2]]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 10 2022 *)

def A165795(n,k):
    if (k==n): return 1
    elif (k==0): return -1
    else: return numerator(1/(n-k)^2 -1/k^2)
flatten([[A165795(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 10 2022

A(n, k) = numerator(1/n^2 - 1/k^2) with A(0,k) = 1 and A(n,0) = -1 (array).
A(n, 0) = -A158388(n).
A(n, k) = A172157(n,k), n>=1.
From G. C. Greubel, Mar 10 2022: (Start)
T(n, k) = numerator(1/(n-k)^2 -1/k^2), with T(n,n) = 1, T(n,0) = -1 (triangle).
A(n, n) = T(2*n, n) = 0^n.
Sum_{k=0..n} T(n, k) = 0^n.
T(n, n-k) = -T(n,k).
T(2*n+1, n) = -A005408(n). (End)

A173651 Triangle T(n,m) read by rows: numerator of 1/(n-m)^2-1/m^2, or -1 if m=0.

Original entry on oeis.org

-1, -1, 0, -1, -3, 3, -1, -8, 0, 8, -1, -15, -5, 5, 15, -1, -24, -3, 0, 3, 24, -1, -35, -21, -7, 7, 21, 35, -1, -48, -2, -16, 0, 16, 2, 48, -1, -63, -45, -1, -9, 9, 1, 45, 63, -1, -80, -15, -40, -5, 0, 5, 40, 15, 80, -1, -99, -77, -55, -33, -11, 11, 33, 55, 77, 99, -1, -120, -6, -8, -3, -24, 0, 24, 3, 8, 6, 120, -1, -143, -117, -91, -65, -39, -13, 13, 39, 65, 91, 117, 143, -1, -168, -35, -112, -21, -56, -7, 0, 7, 56, 21, 112, 35, 168, -1, -195, -165, -5, -105, -3, -5, -15

Offset: 1

Paul Curtz, Nov 24 2010

This is triangle A165507 with an additional column T(n,0)= -1.

The triangle is obtained from the infinite array shown in the comment in A172370 by starting in its row n, column 0 and reading diagonally upwards up to row n=1, column n-1.

			The triangle starts in row n=1 with columns 0<=m<n as
-1;
-1,0,
-1,-3,3,
-1,-8,0,8,
-1,-15,-5,5,15,
-1,-24,-3,0,3,24,
-1,-35,-21,-7,7,21,35,
-1,-48,-2,-16,0,16,2,48,
-1,-63,-45,-1,-9,9,1,45,63,

Cf. A165795.

A173651 := proc(n,m) if m = 0 then -1 ; else 1/(n-m)^2-1/m^2 ; numer(%) ; end if; end proc:

cp's OEIS Frontend

A165795 Array A(n, k) = numerator of 1/n^2 - 1/k^2 with A(0,k) = 1 and A(n,0) = -1, read by antidiagonals.

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