A172370 -id:A172370 - cp's OEIS frontend

A172157 Triangle T(n,m) = numerator of 1/n^2 - 1/m^2, read by rows, with T(n,0) = -1.

-1, -1, -3, -1, -8, -5, -1, -15, -3, -7, -1, -24, -21, -16, -9, -1, -35, -2, -1, -5, -11, -1, -48, -45, -40, -33, -24, -13, -1, -63, -15, -55, -3, -39, -7, -15, -1, -80, -77, -8, -65, -56, -5, -32, -17, -1, -99, -6, -91, -21, -3, -4, -51, -9, -19, -1, -120, -117, -112

Offset: 1

Paul Curtz, Jan 27 2010

The triangle obtained by negating the values of the triangle A120072 and adding a row T(n,0) = -1.

			The full array of numerators starts in row n=1 with columns m>=0 as:
-1...0...3...8..15..24..35..48..63..80..99. A005563
-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
-1.-15..-3..-7...0...9...5..33...3..65..21. A061041
-1.-24.-21.-16..-9...0..11..24..39..56...3. A061043
-1.-35..-2..-1..-5.-11...0..13...7...5...4. A061045
-1.-48.-45.-40.-33.-24.-13...0..15..32..51. A061047
-1.-63.-15.-55..-3.-39..-7.-15...0..17...9. A061049
The triangle is the portion below the main diagonal, left from the zeros, 0<=m<n.

G. C. Greubel, Rows n=1..100 of triangle, flattened

Cf. A172370, A174233, A165795.

T[n_, 0] := -1; T[n_, k_] := 1/n^2 - 1/k^2; Table[Numerator[T[n, k]], {n, 1, 100}, {k, 0, n - 1}] // Flatten (* G. C. Greubel, Sep 19 2018 *)

A174233 Triangle T(n,k) read by rows: the numerator of 1/n^2 - 1/(k-n)^2, 0 <= k < 2n.

Original entry on oeis.org

0, -1, 0, -3, -1, -3, 0, -5, -8, -1, -8, -5, 0, -7, -3, -15, -1, -15, -3, -7, 0, -9, -16, -21, -24, -1, -24, -21, -16, -9, 0, -11, -5, -1, -2, -35, -1, -35, -2, -1, -5, -11, 0, -13, -24, -33, -40, -45, -48, -1, -48, -45, -40, -33, -24, -13, 0, -15, -7, -39, -3, -55, -15, -63

Offset: 1

Paul Curtz, Mar 13 2010

A value of -1 is substituted at the poles where k=n.
The triangle is created by selecting the first 2, 4, 6 etc elements of the array shown in A172370, equivalent to attaching the initial elements of the rows of A172157 to the rows of A174190.
If the first column of zeros is removed from the triangle, each row is left-right symmetric with respect to the center value.

			The triangle starts
  0,  -1;
  0,  -3,  -1,  -3;
  0,  -5,  -8,  -1,  -8,  -5;
  0,  -7,  -3, -15,  -1, -15,  -3,  -7;
  0,  -9, -16, -21, -24,  -1, -24, -21, -16,  -9;
  0, -11,  -5,  -1,  -2, -35,  -1, -35,  -2,  -1,  -5, -11;
  0, -13, -24, -33, -40, -45, -48,  -1, -48, -45, -40, -33, -24, -13;

G. C. Greubel, Rows n=1..100 of triangle, flattened

Cf. A175779, A172370, A061035, A174190, A120072.

A173233 := proc(n,k) if k = n then -1 ; else 1/n^2-1/(k-n)^2 ; numer(%) ; end if; end proc: # R. J. Mathar, Jan 06 2011

T[n_, n_] := -1; T[n_, k_] := 1/n^2 - 1/(k - n)^2; Table[Numerator[T[n, k]], {n, 1, 20}, {k, 0, 2 n - 1}]//Flatten (* G. C. Greubel, Sep 19 2018 *)

A165507 Triangle T(n,m) read by rows: numerator of 1/(1+n-m)^2 - 1/m^2.

Original entry on oeis.org

0, -3, 3, -8, 0, 8, -15, -5, 5, 15, -24, -3, 0, 3, 24, -35, -21, -7, 7, 21, 35, -48, -2, -16, 0, 16, 2, 48, -63, -45, -1, -9, 9, 1, 45, 63, -80, -15, -40, -5, 0, 5, 40, 15, 80, -99, -77, -55, -33, -11, 11, 33, 55, 77, 99, -120, -6, -8, -3, -24, 0, 24, 3, 8, 6, 120

Offset: 1

Paul Curtz, Sep 21 2009

The triangle is obtained from the infinite array shown in the comment in A172370 by starting in column 1 and reading diagonally upwards along increasing columns or starting in column -1 and reading diagonally upwards along decreasing columns.
Equivalence of these two interpretations follows from the mirror symmetry m <-> -m along column m=0 in that array.
T(n,m) is antisymmetric (changes sign) with respect to a central zero if the row index n is odd, and with respect to the separator in the middle of the row if the row index n is even: T(n,m) = -T(n,n+1-m).
An appropriate triangle of denominators is in A143183.

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

G. C. Greubel, Rows n=1..100 of triangle, flattened

Cf. A120072, A143183, A165441.

[[Numerator(1/(n-k+1)^2 - 1/k^2): k in [1..n]]: n in [1..15]]; // G. C. Greubel, Oct 21 2018

A165507 := proc(n,m) 1/(1+n-m)^2-1/m^2 ; numer(%) ; end proc:

Table[Numerator[1/(n-k+1)^2 - 1/k^2], {n,1,15}, {k,1,n}]//Flatten (* G. C. Greubel, Oct 21 2018 *)

for(n=1, 15, for(k=1, n, print1(numerator(1/(n-k+1)^2 - 1/k^2), ", "))) \\ G. C. Greubel, Oct 21 2018

T(n,m) = A173651(1+n,m), m>=1.

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:

A174413 Triangle T(n,m) with the denominator of 1/(n-m)^2-1/n^2, read by rows, 1<=m

Original entry on oeis.org

4, 36, 9, 144, 16, 16, 400, 225, 100, 25, 900, 144, 12, 9, 36, 1764, 1225, 784, 441, 196, 49, 3136, 576, 1600, 64, 576, 64, 64, 5184, 3969, 324, 2025, 1296, 81, 324, 81, 8100, 1600, 4900, 225, 100, 400, 900, 25, 100, 12100, 9801, 7744, 5929, 4356, 3025, 1936, 1089, 484, 121

Offset: 2

Paul Curtz, Mar 19 2010

Obtained by deleting the last entry in each row of A061036 or by reversing rows in A120073.

			Triangle T(n,m) begins:
     4,
    36,    9,
   144,   16,   16,
   400,  225,  100,  25,
   900,  144,   12,   9,  36,
  1764, 1225,  784, 441, 196, 49,
  3136,  576, 1600,  64, 576, 64, 64,

Alois P. Heinz, Table of n, a(n) for n = 2..2017

Cf. A165441, A172370 (numerators).

A174413 := proc(n,m) 1/(n-m)^2-1/n^2 ; denom(%) ; end proc:
seq(seq(A174413(n, k), k=1..n-1), n=2..11); # R. J. Mathar, Jan 27 2011

T[n_, m_] := Denominator[1/(n - m)^2 - 1/n^2];
Table[T[n, m], {n, 2, 11}, {m, 1, n-1}] // Flatten (* Jean-François Alcover, May 18 2018 *)

A174190 Triangle T(n,m) = numerator of 1/n^2-1/(n-m)^2, read by rows.

Original entry on oeis.org

0, 0, -3, 0, -5, -8, 0, -7, -3, -15, 0, -9, -16, -21, -24, 0, -11, -5, -1, -2, -35, 0, -13, -24, -33, -40, -45, -48, 0, -15, -7, -39, -3, -55, -15, -63, 0, -17, -32, -5, -56, -65, -8, -77, -80, 0, -19, -9, -51, -4, -3, -21, -91, -6, -99, 0, -21, -40, -57, -72, -85, -96

Offset: 1

Paul Curtz, Mar 11 2010

The triangle contains the initial values of the array described in A172370.

Ignoring details of column indexing, these are the negated values of A061035.

			The triangle starts in column n=1, rows 0<=m<n as
0;
0,-3;
0,-5,-8;
0,-7,-3,-15;
0,-9,-16,-21,-24;

Comments compactified with reference to A172370, formula and example added - R. J. Mathar, Nov 23 2010

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A172157 Triangle T(n,m) = numerator of 1/n^2 - 1/m^2, read by rows, with T(n,0) = -1.

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A174233 Triangle T(n,k) read by rows: the numerator of 1/n^2 - 1/(k-n)^2, 0 <= k < 2n.

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A165507 Triangle T(n,m) read by rows: numerator of 1/(1+n-m)^2 - 1/m^2.

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

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A174413 Triangle T(n,m) with the denominator of 1/(n-m)^2-1/n^2, read by rows, 1<=m

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A174190 Triangle T(n,m) = numerator of 1/n^2-1/(n-m)^2, read by rows.

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