A120073 -id:A120073 - cp's OEIS frontend

A120077 Denominators of row sums of rational triangle A120072/A120073.

4, 36, 144, 3600, 3600, 176400, 705600, 6350400, 1270080, 153679680, 153679680, 25971865920, 25971865920, 129859329600, 519437318400, 150117385017600, 150117385017600, 54192375991353600, 2167695039654144, 1548353599752960, 221193371393280, 117011293467045120

Offset: 2

Wolfdieter Lang, Jul 20 2006

The first 19 terms coincide with A007407(n), for n>=2. However a(20) = 2167695039654144 and A007407(20) = 10838475198270720 = 5*a(20). Also a(21) = 1548353599752960 and A007407(21) = 221193371393280 = a(21)/7. From n = 22 up to at least n = 100 (checked) both sequences coincide again.
See the W. Lang link under A120072 for more details.
The corresponding numerators are given by A120076.
The n for which a(n) differs from A007407(n) are given by A309829. - Jeppe Stig Nielsen, Aug 18 2019

			The rationals A120076(m)/a(m), m>=2, begin with (3/4, 37/36, 169/144, 4549/3600, 4769/3600, ... ).

Jeppe Stig Nielsen, Table of n, a(n) for n = 2..1150

Cf. A007407, A120070, A120072, A120073, A120074, A120075, A120076, A309829.

A120077:= func< n | Denominator( (&+[1/k^2: k in [1..n]]) -1/n) >;
[A120077(n): n in [2..30]]; // G. C. Greubel, Apr 25 2023

Table[Denominator[HarmonicNumber[n,2] -1/n], {n,2,40}] (* G. C. Greubel, Apr 25 2023 *)

a(n) = denominator(sum(j=1,n-1,1/j^2-1/n^2)) \\ Jeppe Stig Nielsen, Aug 18 2019

a(n) = denominator(sum(j=1,n,1/j^2) - 1/n) \\ Jeppe Stig Nielsen, Aug 18 2019

def A120077(n): return denominator(harmonic_number(n,2) - 1/n)
[A120077(n) for n in range(2,31)] # G. C. Greubel, Apr 25 2023

a(n) = denominator(r(m)), with the rationals r(m) = Sum_{n=1..m-1} A120072(m,n)/A120073(m,n), m >= 2.
The rationals are r(m) = Zeta(2; m-1) - (m-1)/m^2, m>=2, with the partial sums Zeta(2; n) = Sum_{k=1..n} 1/k^2. See the W. Lang link under A103345.
O.g.f. for the rationals r(m), m>=2: log(1-x) + polylog(2,x)/(1-x).

a(21)-a(23) from Jeppe Stig Nielsen, Aug 18 2019

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

Original entry on oeis.org

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.

A120076 Numerators of row sums of rational triangle A120072/A120073.

Original entry on oeis.org

3, 37, 169, 4549, 4769, 241481, 989549, 9072541, 1841321, 225467009, 227698469, 38801207261, 39076419341, 196577627041, 790503882349, 229526961468061, 230480866420061, 83512167402400421, 3351610394325821

Offset: 2

Wolfdieter Lang, Jul 20 2006

The corresponding denominators are given by A120077.

See the W. Lang link under A120072 for more details.

			The rationals a(m)/A120077(m), m>=2, begin with (3/4, 37/36, 169/144, 4549/3600, 4769/3600, ...).

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

Cf. A120070, A120072, A120073, A120074, A120075, A120077.

A120076:= func< n | Numerator( (&+[1/k^2: k in [1..n]]) -1/n) >;
[A120076(n): n in [2..30]]; // G. C. Greubel, Apr 24 2023

Table[Numerator[HarmonicNumber[n,2] -1/n], {n,2,40}] (* G. C. Greubel, Apr 24 2023 *)

def A120076(n): return numerator(harmonic_number(n,2) - 1/n)
[A120076(n) for n in range(2,31)] # G. C. Greubel, Apr 24 2023

a(n) = numerator(r(m)), with the rationals r(m) = Sum_{n=1..m-1} A120072(m,n)/A120073(m,n), m >= 2.
The rationals are r(m) = Zeta(2; m-1) - (m-1)/m^2, m >= 2, with the partial sums Zeta(2; n) = Sum_{k=1..n} 1/k^2. See the W. Lang link in A103345.
O.g.f. for the rationals r(m), m>=2: log(1-x) + polylog(2,x)/(1-x).

A120070 Triangle of numbers used to compute the frequencies of the spectral lines of the hydrogen atom.

Original entry on oeis.org

3, 8, 5, 15, 12, 7, 24, 21, 16, 9, 35, 32, 27, 20, 11, 48, 45, 40, 33, 24, 13, 63, 60, 55, 48, 39, 28, 15, 80, 77, 72, 65, 56, 45, 32, 17, 99, 96, 91, 84, 75, 64, 51, 36, 19, 120, 117, 112, 105, 96, 85, 72, 57, 40, 21

Offset: 2

Wolfdieter Lang, Jul 20 2006

The rationals r(m,n):=a(m,n)/(m^2*n^2), for m-1 >= n, else 0, are used to compute the frequencies of the spectral lines of the H-atom according to quantum theory: nu(m,n) = r(m,n)*c*R' with c*R'=3.287*10^15 s^(-1) an approximation for the Rydberg frequency. R' indicates, that the correction factor 1/(1+m_e/m_p), approximately 0.9995, with the masses for the electron and proton, has been used for the Rydberg constant R_infinity. c:=299792458 m/s is, per definition, the velocity of light in vacuo (see A003678).
In order to compute the wave length of the spectral lines approximately one uses the reciprocal rationals: lambda(m,n):= c/nu(m,n) = (1/r(m,n))*91.1961 nm. 1 nm = 10^{-9} m. For the corresponding energies one uses approximately E(m,n) = r(m,n)*13.599 eV (electron Volts).
The author was inspired by Dewdney's book to compile this table and related ones.
For the approximate frequencies, energies and wavelengths of the first members of the Lyman (n=1, m>=2), Balmer (n=2, m>=3), Paschen (n=3, m>=4), Brackett (n=4, m>=5) and Pfund (n=5, m>=6) series see the W. Lang link under A120072.
Frenicle wrote this as a(n+1) = A140978(n) - A133819(n-1). - Paul Curtz, Aug 19 2008
This triangle also has an interpretation related to particle spin. For proper offset such that T(0,0) = 3, then, where h-bar = h/(2*Pi) = A003676/A019692 (= The Dirac constant, also known as Planck's reduced constant) and Spin(n/2) = h-bar/2*sqrt(n(n+2)), it follows that: h-bar/2*sqrt(T(r,k)) = h-bar/2*sqrt(T(r,0) - T(k-1,0)) = sqrt((Spin((r+1)/2))^2 - (Spin(k/2))^2). For example, for r = k = 4, then h-bar/2*sqrt(11) = h-bar/2*sqrt(T(4,4)) = h-bar/2*sqrt(T(4,0) - T(3,0)) = sqrt(h-bar^2/4*T(4,0) - h-bar^2/4*T(3,0)) = sqrt(h-bar^2/4*35 - h-bar^2/4*24) = sqrt((Spin((4+1)/2))^2 - (Spin(4/2))^2); 35 = 5*(5+2) & 24 = 4*(4+2). - Raphie Frank, Dec 30 2012

			Triangle begins
  [ 3];
  [ 8, 5];
  [15, 12,  7];
  [24, 21, 16,  9];
  [35, 32, 27, 20, 11];
  ...

A. K. Dewdney, Reise in das Innere der Mathematik, Birkhäuser, Basel, 2000, pp. 148-154; engl.: A Mathematical Mystery Tour, John Wiley & Sons, N.Y., 1999.

Stanislav Sykora, Table of n, a(n) for n = 2..79801
W. Lang: First ten rows and more.
M. de Frenicle, Methode pour trouver la solutions des problemes par les exclusions, in: Divers ouvrages des mathematiques et de physique par messieurs de l'academie royale des sciences, (1693) pp 1-44, page 11.
Wikipedia, Spin (physics)
Wikipedia, Hydrogen spectral series

Row sums give A016061(n-1), n>=2.

Cf. A120072/A120073 numerator and denominator tables for rationals r(m, n).

ColumnForm[Table[n^2 - k^2, {n, 2, 13}, {k, n - 1}], Center] (* Alonso del Arte, Oct 26 2011 *)

nmax=400;a=vector(1+nmax*(nmax-1)\2);idx=1;for(n=2,nmax,for(k=1,n-1,a[idx]=n*n-k*k;idx++)) \\ Stanislav Sykora, Feb 17 2014

T(n,k)=n^2-k^2;
for (n=1,10,for(k=1,n-1, print1(T(n,k),", ")));
\\ Joerg Arndt, Feb 23 2014

a(m,n) = m^2 - n^2 for m-1 >= n, otherwise 0.
G.f. for column n=1,2,...: x^(n+1)*((2*n+1)- (2*n-1)*x)/(1-x)^3.
G.f. for rationals r(m,n), n=1,2,...,10 see W. Lang link.
T(r,k) = T(r,0) - T(k-1,0), T(0,0) = 3. - Raphie Frank, Dec 27 2012

A120072 Numerator triangle for hydrogen spectrum rationals.

Original entry on oeis.org

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

Offset: 2

Wolfdieter Lang, Jul 20 2006

Frequencies or energies of the spectral lines of the hydrogen (H) atom are given, according to quantum theory, by r(m,n)*3.287*PHz (1 Peta Hertz= 10^15 s^{-1}) or r(m,n)*13.599 eV (electron Volts), respectively. The wave lengths are lambda(m,n) = (1/r(m,n))* 91.196 nm (all decimals rounded). See the W. Lang link for more details.
The spectral series for n=1,2,...,7, m>=n+1, are named after Lyman, Balmer, Paschen, Brackett, Pfund, Humphreys, Hansen-Strong, respectively.
The corresponding denominator triangle is A120073.
The rationals are r(m,n):= a(m,n)/A120073(m,n) = A120070(m,n)/(m^2*n^2) = 1/ n^2 - 1/m^2 and they are given in lowest terms.

			For the rational triangle see W. Lang link.
Numerator triangle begins as:
   3;
   8,  5;
  15,  3,  7;
  24, 21, 16,  9;
  35,  2,  1,  5, 11;
  48, 45, 40, 33, 24, 13;
  63, 15, 55,  3, 39,  7, 15;
  80, 77,  8, 65, 56,  5, 32, 17;
  99,  6, 91, 21,  3,  4, 51,  9, 19;

G. C. Greubel, Rows n = 2..50 of the triangle, flattened
Wolfdieter Lang, First ten rows, rationals and more.
T. Lyman, The Spectrum of Hydrogen in the Region of Extremely Short Wave-Lengths, The Astrophysical Journal, 23 (April 1906), 181-210. - _Paul Curtz_, May 30 2017

Row sums give A120074.
Row sums of r(m, n) triangle give A120076(m)/A120077(m), m>=2.
Cf. A120070, A120073, A120075, A126252.

[Numerator(1/k^2 - 1/n^2): k in [1..n-1], n in [2..18]]; // G. C. Greubel, Apr 24 2023

Table[1/n^2 - 1/m^2, {m,2,12}, {n,m-1}]//Flatten//Numerator (* Jean-François Alcover, Sep 16 2013 *)

def A120072(n,k): return numerator(1/k^2 - 1/n^2)
flatten([[A120072(n,k) for k in range(1,n)] for n in range(2,19)]) # G. C. Greubel, Apr 24 2023

a(m,n) = numerator(r(m,n)) with r(m,n) = 1/n^2 - 1/m^2, m>=2, n=1..m-1.

The g.f.s for the columns n=1,..,10 of triangle r(m,n) = a(m, n) / A120073(m, n), m >= 2, 1 <= n <= m-1, are given in the W. Lang link.

A119937 Triangle of numbers related to the spectrum of the hydrogen (H) atom.

Original entry on oeis.org

3, 32, 5, 135, 27, 7, 3456, 756, 256, 81, 3500, 800, 300, 125, 44, 172800, 40500, 16000, 7425, 3456, 1300, 694575, 165375, 67375, 33075, 17199, 8575, 3375, 6272000, 1509200, 627200, 318500, 175616, 98000, 51200

Offset: 2

Wolfdieter Lang, Jul 20 2006

The rational number triangle r(m,n):=A120072(m,n)/A120073(m,n), used to compute the spectral series of the hydrogen atom, is mapped to this nonnegative number triangle by multiplying the least common multiples (LCM) for each row m.

			[3]; [32,5]; [135,27,7]; [3456,756,256,81]; [3500,800,300,125,44]; ...

W. Lang: First ten rows.

The LCM sequence which has been used here is [4, 36, 144, 3600, 3600, 176400, 705600, 6350400, 6350400, 768398400, ...] = A051418(m) = (A003418(m))^2 = (2*A025555(m-1))^2, m >= 2.

The row sums give A119938.

a(m,n) = r(m,n)*lcm_{k=1..m-1} seq(r(m,k)) with r(m,n) = 1/n^2 - 1/m^2 = A120072(m,n)/A120073(m,n), m >= 2, n = 1..m-1.

A120074 Row sums of triangle A120072 (numerator triangle for H atom spectrum).

Original entry on oeis.org

3, 13, 25, 70, 54, 203, 197, 340, 303, 825, 445, 1378, 892, 1221, 1565, 3128, 1545, 4389, 2427, 3592, 3688, 7843, 3589, 8420, 6191, 9097, 7135, 15834, 5774, 19375, 12493, 14814, 14147, 19647, 12264, 33078

Offset: 2

Wolfdieter Lang, Jul 20 2006

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

Cf. A120070, A120072, A120073, A120075, A120076, A120077.

A120072:= func< n,k | Numerator(1/k^2 - 1/n^2) >;
[(&+[A120072(n,k): k in [1..n-1]]): n in [2..50]]; // G. C. Greubel, Apr 24 2023

Table[Sum[1/n^2 - 1/m^2 //Numerator, {n,m-1}], {m,2,40}]  (* Jean-François Alcover, Sep 16 2013 *)

def A120072(n,k): return numerator(1/k^2 - 1/n^2)
[sum(A120072(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} A120072(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

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

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 *)

cp's OEIS Frontend

A120077 Denominators of row sums of rational triangle A120072/A120073.

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

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A120076 Numerators of row sums of rational triangle A120072/A120073.

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A120070 Triangle of numbers used to compute the frequencies of the spectral lines of the hydrogen atom.

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A120072 Numerator triangle for hydrogen spectrum rationals.

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A119937 Triangle of numbers related to the spectrum of the hydrogen (H) atom.

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A120074 Row sums of triangle A120072 (numerator triangle for H atom spectrum).

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