A172157 -id:A172157 - cp's OEIS frontend

A172370 Mirrored triangle A120072 read by rows.

3, 5, 8, 7, 3, 15, 9, 16, 21, 24, 11, 5, 1, 2, 35, 13, 24, 33, 40, 45, 48, 15, 7, 39, 3, 55, 15, 63, 17, 32, 5, 56, 65, 8, 77, 80, 19, 9, 51, 4, 3, 21, 91, 6, 99, 21, 40, 57, 72, 85, 96, 105, 112, 117, 120, 23, 11, 7, 5, 95, 1, 119, 1, 5, 35, 143, 25, 48, 69, 88, 105, 120, 133, 144

Offset: 2

Paul Curtz, Feb 01 2010

A table of numerators of 1/n^2 - 1/m^2 extended to negative m looks as follows, stacked such that values of common m are aligned
and the central column of -1 is defined for m=0:
.............................0..-1...0...3...8..15..24..35..48..63..80..99. A005563
.........................0..-3..-1..-3...0...5...3..21...2..45..15..77...6. A061037
.....................0..-5..-8..-1..-8..-5...0...7..16...1..40..55...8..91. A061039
.................0..-7..-3.-15..-1.-15..-3..-7...0...9...5..33...3..65..21. A061041
.............0..-9.-16.-21.-24..-1.-24.-21.-16..-9...0..11..24..39..56...3. A061043
.........0.-11..-5..-1..-2.-35..-1.-35..-2..-1..-5.-11...0..13...7...5...4. A061045
.....0.-13.-24.-33.-40.-45.-48..-1.-48.-45.-40.-33.-24.-13...0..15..32..51. A061047
.0.-15..-7.-39..-3.-55.-15.-63..-1.-63.-15.-55..-3.-39..-7.-15...0..17...9. A061049
The row-reversed variant of A120072 appears (negated) after the leftmost 0.
Equals A061035 with the first column removed. - Georg Fischer, Jul 26 2023

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

G. C. Greubel, Rows n = 2..100 of triangle, flattened

Lower diagonal gives: A070262, A061037(n+2).

Cf. A061035, A172157, A165795.

[[Numerator(1/(n-k)^2 -1/n^2): k in [1..n-1]]: n in [2..20]]; // G. C. Greubel, Sep 20 2018

Table[Numerator[1/(n-k)^2 -1/n^2], {n, 2, 20}, {k, 1, n-1}]//Flatten (* G. C. Greubel, Sep 20 2018 *)

for(n=2,20, for(k=1,n-1, print1(numerator(1/(n-k)^2 -1/n^2), ", "))) \\ G. C. Greubel, Sep 20 2018

T(n,m) = numerator of 1/(n-m)^2 - 1/n^2, n >= 2, 1 <= m < n. - R. J. Mathar, Nov 23 2010

Comment rewritten and offset set to 2 by R. J. Mathar, Nov 23 2010

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.

Original entry on oeis.org

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)

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 *)

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 *)

A174381 Triangle for denominators of half extended Rydberg-Ritz spectrum of the hydrogenic spectra. a(n) is an antidiagonal writing of array in A171522 without first column.

Original entry on oeis.org

0, 0, 4, 0, 9, 36, 0, 16, 16, 144, 0, 25, 100, 225, 400, 0, 36, 9, 12, 144, 900, 0, 49, 196, 441, 784, 1225, 1764, 0, 64, 64, 576, 64, 1600, 576, 3136, 0, 81, 324, 81, 1296, 2025, 324, 3969, 5184, 0, 100, 25, 900, 400, 100, 225, 4900, 1600, 8100, 0, 121, 484, 1089

Offset: 0

Paul Curtz, Mar 17 2010

Companion to A172157 (numerators). Hence -1/0; -1/0,-3/4; -1/0,-8/9,-5/36; -1/0,-15/16,-3/16,-7/144; -1/0,-24/25,-21/100,-16/225,-9/400; for 1) (-1/0, A005563/A000290(n+1))=A067998(n+1)/A000290 Lyman; 2) -1/0, -3/4, A061037/A061038 Balmer ; 3) -1/0, -8/9, -5/36, A061039/A061040)=A171709(n+3)/ Paschen; 4) (-1/0, -15/16, -3/16, -7/144, A061041/A061042 Brackett; .. .

a(n)= 0, (mix 0 or A000004 , n-th row of A120073)

cp's OEIS Frontend

A172370 Mirrored triangle A120072 read by rows.

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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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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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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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A174381 Triangle for denominators of half extended Rydberg-Ritz spectrum of the hydrogenic spectra. a(n) is an antidiagonal writing of array in A171522 without first column.

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