A174233 -id:A174233 - 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 *)

A174680 Numerator of 1/16 - 1/n^2, using -1 at the pole where n=0.

Original entry on oeis.org

-1, -15, -3, -7, 0, 9, 5, 33, 3, 65, 21, 105, 1, 153, 45, 209, 15, 273, 77, 345, 3, 425, 117, 513, 35, 609, 165, 713, 3, 825, 221, 945, 63, 1073, 285, 1209, 5, 1353, 357, 1505, 99, 1665, 437, 1833, 15, 2009, 525, 2193, 143, 2385, 621, 2585, 21, 2793, 725, 3009, 195, 3233, 837, 3465, 14, 3705

Offset: 0

Paul Curtz, Nov 30 2010

Extends the Brackett spectrum to negative principal quantum numbers in the spirit of A144477 and A171709.

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. A174233, A067998.

Table[(n^2 - 16)/(GCD[n^2 - 16, 16*n^2]), {n, 0, 100}] (* G. C. Greubel, Sep 16 2018 *)

{a(n) = (n^2 - 16) / gcd(n^2 - 16, 16 * n^2)}; /* Michael Somos, Jan 06 2011 */

a(n) = A061041(n), n >= 4.
a(n) = A172157(4,n), n >= 1.
a(n) = a(-n) for all n in Z.

removed a(-4)-a(-1) since a(-n)=a(n) by Michael Somos, Jan 06 2011

A175779 Triangle T(n,m) read by rows: numerator of 1/(n-m)^2 - 1/n^2.

Original entry on oeis.org

1, 0, 1, 0, 3, 1, 0, 5, 8, 1, 0, 7, 3, 15, 1, 0, 9, 16, 21, 24, 1, 0, 11, 5, 1, 2, 35, 1, 0, 13, 24, 33, 40, 45, 48, 1, 0, 15, 7, 39, 3, 55, 15, 63, 1, 0, 17, 32, 5, 56, 65, 8, 77, 80

Offset: 0

Paul Curtz, Dec 04 2010

T(n,0) is set to zero at the pole m=0. T(n,n) is otherwise set to 1 at the pole n=m.
This is the triangle A061035 augmented by a diagonal of 1's.
Essentially the same information is in A120072, A166492, A172157 and A174233.

			The triangle starts in row n=0 with columns 0<=m<=n as:
.1.
.0..1.
.0..3..1.
.0..5..8..1.
.0..7..3.15..1.
.0..9.16.21.24..1.
.0.11..5..1..2.35..1.
.0.13.24.33.40.45.48..1.
.0.15..7.39..3.55.15.63..1.
.0.17.32..5.56.65..8.77.80..1.
.0.19..9.51..4..3.21.91..6.99..1.

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

Cf. A172157, A166925, A171522 (denominators)

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

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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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A174680 Numerator of 1/16 - 1/n^2, using -1 at the pole where n=0.

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A175779 Triangle T(n,m) read by rows: numerator of 1/(n-m)^2 - 1/n^2.

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