A120072 -id:A120072 - cp's OEIS frontend

A120075 Row sums of triangle A120073 (denominator triangle for H atom spectrum).

4, 45, 176, 750, 1101, 4459, 6080, 13284, 16350, 46585, 33954, 109850, 92463, 142705, 198400, 432344, 255096, 761349, 500355, 824866, 925529, 2007555, 1044616, 2612500, 2158130, 3301641, 2848741

Offset: 2

Wolfdieter Lang, Jul 20 2006

G. C. Greubel, Table of n, a(n) for n = 2..1000

Cf. A120070, A120072, A120073, A120074, A120076, A120077.

A120073:= func< n,k | Denominator(1/k^2 - 1/n^2) >;
[(&+[A120073(n,k): k in [1..n-1]]): n in [2..50]]; // G. C. Greubel, Apr 24 2023

A120075[n_]:= Sum[Denominator[1/k^2 -1/n^2], {k,n-1}];
Table[A120075[n], {n,2,50}] (* G. C. Greubel, Apr 24 2023 *)

def A120073(n,k): return denominator(1/k^2 - 1/n^2)
[sum(A120073(n,k) for k in range(1,n)) for n in range(2,51)] # G. C. Greubel, Apr 24 2023

a(n) = Sum_{k=1..n-1} A120073(n,k), for n >= 2.

A126252 Wavenumbers of red, turquoise, blue, indigo and violet in the spectrum of hydrogen, as first measured by Robert Bunsen and Gustav Kirchhoff in 1859.

Original entry on oeis.org

1523310, 2056410, 2303240, 2437290, 2518130

Offset: 1

Reinhard Zumkeller, Dec 22 2006

How Johann Jakob Ballmer found his formula in 1885 by analyzing and manipulating the ratios of these data:
r(1) = a(1)/a(1) = 1,
a(2)/a(1) = 1.349961..., rounded: r(2) = 135/100 = 27/20,
a(3)/a(1) = 1.511996..., rounded: r(3) = 1512/1000 = 189/125,
a(4)/a(1) = 1.599996..., rounded: r(4) = 16/10 = 8/5,
a(5)/a(1) = 1.6530647..., r(5) = 81/49 = 2-1/(3-1/(9-1/2)), derived from a(5)/a(1) = 2-1/(3-1/(9-3095/6216)) when replacing 3095/6216 by 1/2;
the multiplication of these fractions by 5/36 is the key trick to get more handy figures to see eventually increasing squares in the denominators by an appropriate expansion:
b(1) = r(1)*5/36 = 5 / 36,
b(2) = r(2)*5/36 = 3 / 16,
b(3) = r(3)*5/36 = 21 / 100,
b(4) = r(4)*5/36 = 2 / 9,
b(5) = r(5)*5/36 = 45 / 196;
... b(1) .|.... b(2) ..|.... b(3) ..|.... b(4) ..|.... b(5),
... 5/36 .|.... 3/16 ..|... 21/100 .|.... 2/9 ...|... 45/196,
... 5/36 .|... 12/64 ..|... 21/100 .|... 32/144 .|... 45/196,
(9-4)/9*4 |(16-4)/16*4 |(25-4)/25*4 |(36-4)/36*4 |(49-4)/49*4,
this last step was the crowning achievement: the discovery of the pattern (x-y)/x*y,
b(n) = ((n+2)^2 - 4)/(4*(n+2)^2) = 1/4 - 1/(n+2)^2;
1<=n<=5: b(n) = A061037(n+2)/A061038(n+2) = A120072(n+2,2)/A120073(n+2,2).

R. Taschner, Der Zahlen gigantischer Schatten, Vieweg 2005, 137-143.

Science Trek, Balmer Formula
Eric Weisstein's World of Physics, Balmer Formula
Wikipedia, Empirical formula
Wikipedia, Johann Jakob Balmer
Wikipedia, Robert Wilhelm Bunsen
Wikipedia, Gustav Robert Kirchhoff

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.

A120078 Coefficient triangle of numerator polynomials appearing in certain column o.g.f.s related to the H-atom spectrum.

Original entry on oeis.org

1, 4, -3, 36, -27, -5, 144, -108, -20, -7, 3600, -2700, -500, -175, -81, 3600, -2700, -500, -175, -81, -44, 176400, -132300, -24500, -8575, -3969, -2156, -1300, 705600, -529200, -98000, -34300, -15876, -8624, -5200, -3375, 6350400, -4762800, -882000, -308700, -142884, -77616, -46800, -30375, -20825

Offset: 1

Wolfdieter Lang, Jul 20 2006

The row polynomials P(n,x) = Sum_{k=1..n-1} a(n,k)*x^k, n >= 1, appear in the numerator of the o.g.f. for column n of the triangle of rationals A120072(m,n)/A120073(m,n), m >= 2, n = 1..m-1. P(n,x) has degree n-1.

See the W. Lang link under A120072 for the precise form of the o.g.f.s: G(x,n) = -dilog(1-x) + x*P(n,4)/*(A(n)*(n^2)*(1-x)), with A(n) = [1, 1, 4, 9, 144, 100, 3600, 11025, 78400, 63504, ...] = conjectured to be A027451(n), n >= 1.

			For n=2 the o.g.f. of A120072(m,2)/A120073(m,2) (=[5/36, 3/16, 21/100, 2/9, ...]) is G(x,2) = -dilog(1-x) + x*P(2,x)/(1*4*(1-x)) = -dilog(1-x) + x*(4-3*x)/(4*(1-x)).
Triangle begins:
       1;
       4,      -3;
      36,     -27,     -5;
     144,    -108,    -20,    -7;
    3600,   -2700,   -500,  -175,   -81;
    3600,   -2700,   -500,  -175,   -81,   -44;
  176400, -132300, -24500, -8575, -3969, -2156, -1300;

G. C. Greubel, Rows n = 1..50 of the triangle, flattened
Wolfdieter Lang, First ten rows

Row sums (unsigned) give A120079.
Signed row sums conjectured to coincide with A027451.
Cf. A027451, A051418, A120072, A120073, A120079.

f:= func< n | n eq 1 select 1 else 1/n^2 -1/(n-1)^2 >;
A120078:= func< n,k | (Lcm([1..n]))^2*f(k) >;
[A120078(n,k): k in [1..n], n in [1..15]]; // G. C. Greubel, Apr 26 2023

Table[(Apply[LCM, Range[n]])^2*If[k==1, 1, (1-2*k)/(k*(k-1))^2], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Apr 26 2023 *)

def f(k): return 1 if (k==1) else 1/k^2 - 1/(k-1)^2
def A120078(n,k): return (lcm(range(1, n+1)))^2*f(k)
flatten([[A120078(n,k) for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Apr 26 2023

T(n, k) = A051418(n) * (1 if k = 1 otherwise 1/k^2 - 1/(k-1)^2). - G. C. Greubel, Apr 26 2023

A166492 Table of numerators of A120070(n,m)/A002260(n,m), 1 <= m < n.

Original entry on oeis.org

3, 8, 5, 15, 6, 7, 24, 21, 16, 9, 35, 16, 9, 5, 11, 48, 45, 40, 33, 24, 13, 63, 30, 55, 12, 39, 14, 15, 80, 77, 24, 65, 56, 15, 32, 17, 99, 48, 91, 21, 15, 32, 51, 9, 19, 120, 117, 112, 105, 96, 85, 72, 57, 40, 21, 143, 70, 45, 32, 119, 18, 95, 10, 7, 22, 23, 168, 165, 160, 153, 144

Offset: 2

Paul Curtz, Oct 15 2009

Numerators of the fractions (n+m)*(n-m)/m.
The numerical values are between A120070(n,m) and A120072(n,m), see A164561.
If we "flatten" the table (enumerate the sequence starting at 1 instead of using the double index), the positions where a common factor is removed from the numerator A120070 and denominator A002260 are at 5, 12, 13, etc., as given by A076537.

Cf. A129326.

A120070 := proc(n,m) (n+m)*(n-m) ; end:
A002260 := proc(n,m) m ; end:
A166492 := proc(n,m) A120070(n,m)/A002260(n+1,m) ; numer(%) ; end: seq(seq(A166492(n,m),m=1..n-1),n=2..17) ; # R. J. Mathar, Oct 21 2009

Edited and extended by R. J. Mathar, Oct 21 2009

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 *)

A309829 Numbers k for which A120077(k) != A007407(k).

Original entry on oeis.org

20, 21, 110, 136, 156, 930, 44310

Offset: 1

Jeppe Stig Nielsen, Aug 18 2019

The sequence of rationals related to A120077 is f(k) = Sum_{j=1..k-1} (1/j^2 - 1/k^2), motivated by each term's interpretation as the energy difference between shells k and j in a hydrogen atom model. This can easily be seen to be equal to f(k) = (Sum_{j=1..k} 1/j^2) - 1/k. Compare this with g(k) = Sum_{j=1..k} 1/j^2 which is the starting point for A007407. The question is, when does the final subtraction of 1/k change the denominator (in lowest term)? In one case (k=21), the denominator belonging to f(k) is greater than that belonging to g(k). In cases k=20, 110, 136, 156, 930, 44310, the opposite is true.
Will gcd(A120077(k), A007407(k)) always be one of the numbers A120077(k) and A007407(k)?
Should this sequence be infinite?

Cf. A120072, A120077, A007407.

s=1; for(n=2, +oo, s += 1/n^2; denominator(s)!=denominator(s-1/n) && print1(n, ", "))

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A120075 Row sums of triangle A120073 (denominator triangle for H atom spectrum).

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A126252 Wavenumbers of red, turquoise, blue, indigo and violet in the spectrum of hydrogen, as first measured by Robert Bunsen and Gustav Kirchhoff in 1859.

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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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A120078 Coefficient triangle of numerator polynomials appearing in certain column o.g.f.s related to the H-atom spectrum.

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A166492 Table of numerators of A120070(n,m)/A002260(n,m), 1 <= m < n.

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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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A309829 Numbers k for which A120077(k) != A007407(k).

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