Triangle of rationals r(m,n):= A120072(m,n)/A120073(m,n), m>=2, 1<= n <= m-1.  

  r(m,n) = A120070(m,n)/(m^2*n^2) = 1/n^2 - 1/m^2. 

  Used for spectrum of H-atom (see below).



    m\n      1       2         3        4        5           6         7          8        9       10 ...
 

    2       3/4      0         0        0        0           0         0         0         0        0
 
    3       8/9     5/36       0        0        0           0         0         0         0        0 
 
    4      15/16    3/16     7/144      0        0           0         0         0         0        0

    5      24/25   21/100   16/225    9/400      0           0         0         0         0        0
 
    6      35/36    2/9      1/12     5/144     11/900       0         0         0         0        0

    7      48/49   45/196   40/441    33/784    24/1225    13/1764     0         0         0        0

    8      63/64   15/64    55/576     3/64     39/1600    7/576     15/3136     0         0        0 

    9      80/81   77/324    8/81     65/1296   56/2025    5/324     32/3969   17/5184     0        0
    
   10     99/100    6/25    91/900    21/400    3/100      4/225     51/4900   9/1600  19/8100      0
 
   11    120/121  117/484  112/1089  105/1936  96/3025    85/4356    72/5929   57/7744  40/9801  21/12100

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 Row sums give the rationals r(m):=A120076(m)/A120077(m), m>=2:

 [3/4, 37/36, 169/144, 4549/3600, 4769/3600, 241481/176400, 989549/705600, 9072541/6350400, 1841321/1270080, 225467009/153679680,...].

  r(m)= Zeta(2;m-1) - (m-1)/m^2, m>=2, with the partial sums Zeta(2;n):=sum(1/k^2,k=1..n). See the W.Lang link under A103345.
  
  O.g.f. for r(m), m>=2:  R(x):= ln(1-x) + polylog(2,x)/(1-x) = ln(1-x) + (dilog(1-x))/(1-x).
 
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 The generating function for the column n numbers, R(n,x):= sum(r(m,n)*x^m,m=n+1..infinity), n>=1, has the following form:

 R(n,x) = -dilog(1-x) + x*P(n,x))/(A(n)*(n^2)*(1-x)), n>=1, where dilog(1-x)=polylog(2,x), 

 the sequence A(n)=[1,1,4,9,144,100,3600,11025,78400,63504,...] =  A027451(n) (conjecture), n>=1. See below for more on A(n).  

 and the first ten  polynomials P(n,x) are (see the coefficient triangle A120078):

  n=1:   P(1,x) = 1,  

  n=2:   P(2,x) = 4-3*x,

  n=3:   P(3,x) = 36-27*x-5*x^2,

  n=4:   P(4,x) = 144-108*x-20*x^2-7*x^3,

  n=5:   P(5,x) = 3600-2700*x-500*x^2-175*x^3-81*x^4,

  n=6:   P(6,x) = 3600-2700*x-500*x^2-175*x^3-81*x^4-44*x^5,

  n=7:   P(7,x) = 176400-132300*x-24500*x^2-8575*x^3-3969*x^4-2156*x^5-1300*x^6,

  n=8:   P(8,x) = 705600-529200*x-98000*x^2-34300*x^3-15876*x^4-8624*x^5-5200*x^6-3375*x^7,

  n=9:   P(9,x) = 6350400-4762800*x-882000*x^2-308700*x^3-142884*x^4-77616*x^5-46800*x^6-30375*x^7-20825*x^8,

  n=10: P(10,x) = 6350400-4762800*x-882000*x^2-308700*x^3-142884*x^4-77616*x^5-46800*x^6-30375*x^7-20825*x^8-14896*x^9

  ...

  These integer polynomials P(n,x) of degree n-1 can be expessed in terms of partial sums of the k=2 polylog defined by

  polylog(2;x,n):=sum((x^m)/m^2,m=1..n), n>=1, in the follwing way.
  
  P(n,x)/A(n) = x^n + n^2*(1-x)*polylogknx(2,n,x)/x, with the numbers A(n) from above. 

  If one uses x^n + n^2*(1-x)*polylogknx(2,n,x)/x = (n^2)*(1 - sum(((2*k+1)/(k*(k+1)))^2)*x^k,k=1..n-1), n>=1, with the sum 

  put to zero for n=1, it becomes clear that P(n,x) becomes an integer polynomial if one takes 

  A(n)=LCM((1*2)^2, (2*3)^2,...,(n-2)*(n-1)^2, (n-1)^2) = LCM(seq((k*(k+1)))^2,k=1..n-1)) = (LCM(seq(k*(k+1)),k=1..n-1))^2, n>=2.  A(1):=1.

  Therefore A(n)=[1,1,4,36,144,3600,3600,176400,705600,6350400,...],n>=1,  the squares of [1,1,2,6,12,60,60,420,840,2520,...] = A003418(n-1).
  
  This follows from the fact that (2*k+1) (odd) never divides (k*(k+1))^2 (even), and only the last term in the sum is divisible by n^2. 
  
  Therefore, A(n) is the least common multiple (LCM) of the denominators of the entries in  row number n, n>=1, of the rational triangle 

  whose row polynomials are 
 
  p(n,x):= x^n + n^2*(1-x)*polylogknx(2,n,x)/x = (n^2)*(1 - sum(((2*k+1)/(k*(k+1)))^2)*x^k,k=1..n-1), n>=1, with the sum put to zero for n=1.
  
 
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 The triangle for the numerators is A120072(m,n):

 a(m,n) tabl head (triangle) for  A120072


    m\n      1      2      3      4     5     6     7     8     9    10 ...


    2        3      0      0      0     0     0     0     0     0     0

    3        8      5      0      0     0     0     0     0     0     0

    4       15      3      7      0     0     0     0     0     0     0

    5       24     21     16      9     0     0     0     0     0     0

    6       35      2      1      5    11     0     0     0     0     0

    7       48     45     40     33    24    13     0     0     0     0

    8       63     15     55      3    39     7    15     0     0     0

    9       80     77      8     65    56     5    32    17     0     0

   10       99      6     91     21     3     4    51     9    19     0

   11      120    117    112    105    96    85    72    57    40    21
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 The row sums give  [3, 13, 25, 70, 54, 203, 197, 340, 303, 825, ...] = A120074(m), m>=2.

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 The triangle for the denominators is A120073(m,n):

 a(m,n) tabl head (triangle) for  A120073


    m\n      1      2       3       4       5       6       7       8       9       10 ...

 
    2        4      0       0       0       0       0       0       0       0        0
 
    3        9     36       0       0       0       0       0       0       0        0

    4       16     16     144       0       0       0       0       0       0        0

    5       25    100     225     400       0       0       0       0       0        0

    6       36      9      12     144     900       0       0       0       0        0

    7       49    196     441     784    1225    1764       0       0       0        0

    8       64     64     576      64    1600     576    3136       0       0        0

    9       81    324      81    1296    2025     324    3969    5184       0        0

   10      100     25     900     400     100     225    4900    1600    8100        0

   11      121    484    1089    1936    3025    4356    5929    7744    9801    12100

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  The row sums give: [4, 45, 176, 750, 1101, 4459, 6080, 13284, 16350, 46585,....] = A120075(m), m>=2.

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 The frequencies of the spectral lines of the  H-atom  are (approximately):  

 nu(m,n) = r(m,n) * 3.287*10^{15} s^{-1} = r(m,n) * 3.287 PHz,

 P=Peta for 10^15, Hz=Hertz=1/s.

  
 The corresponding energies:  E(m,n) = r(m,n)*13.599 eV,

 (eV= electron volts, 1 eV = 1.602 176 53(14)*10^{-19} J , J=Joule).  

 
 The corresponding wave lengths:  lambda(m,n) = (1/r(m,n)) * 91.196 nm,

 (1 nm = 10^{-9} m)

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 The n=1,2,3,4 and 5 series, for m>=2,3,4,5 and 6, bear the names of Lyman, Balmer, Paschen, Brackett and Pfund, 

 respectively.

 e.g.: n=2 (Balmer series) partly in the visible range (approximate values):

 lambda(3,2) = 657 nm (red), lambda(4,2) = 486 nm (green),  lambda(5,2) = 434 nm (blue), lambda(6,2) = 410 nm (violet), 

 lambda(7,2) = 397 nm (violet),  lambda(8,2) = 389 nm (violet), lambda(9,2) = 384 nm (ultraviolet), ...

 
 The corresponding (approximate) frequencies are (T=Tera = 10^{12}):

  nu(3,2) = 457 THz, nu(4,2) = 617*THz,  nu(5,2) = 731*THz, nu(6,2) =755*THz,  

  nu(7,2) = 771*THz, nu(8,2) = 782*THz,  nu(9,2) = 790*THz,  ...

 The corresponding enrgies are (4 digits):

 1.89*eV, 2.55*eV, 2.86*eV, 3.02*eV, 3.12*eV, 3.19*eV, 3.23*eV, ...

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 Lyman series (n=1, m>=2) in the ultraviolet: 

 frequencies (3 digits):

 2.47*PHz, 2.92*PHz, 3.08*PHz, 3.16*PHz, 3.20*PHz, 3.22*PHz, 3.24*PHz, 3.25*PHz, 3.26*PHz, 3.26*PHz, 3.27*PHz, ...

 wavelengths (4 digits): 

 121.6*nm, 102.6*nm, 97.28*nm, 95.00*nm, 93.81*nm, 93.10*nm, 92.65*nm, 92.34*nm, 92.12*nm, 91.96*nm, 91.84*nm,... 

 energies (4 digits):

 10.20*eV, 12.09*eV, 12.75*eV, 13.06*eV, 13.22*eV, 13.32*eV, 13.39*eV, 13.43*eV, 13.46*eV, 13.49*eV, 13.51*eV, ...


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 Paschen series (n=3, m>=4) in the infrared:

 frequencies (3 digits):

 160 THz, 234 THz, 274 THz, 298 THz, 314 THz, 325 THz, 333 THz, 338 THz, 343 THz, 346 THz, ...

 wavelengths (4 digits): 

 1876.*nm, 1282.*nm, 1094.*nm, 1005.*nm, 955.1*nm, 923.4*nm, 902.0*nm, 886.8*nm, 875.5*nm, 867.0*nm,...

 energies (3 digits):
 
 .661*eV, .967*eV, 1.13*eV, 1.23*eV, 1.30*eV, 1.34*eV, 1.37*eV, 1.40*eV, 1.42*eV, 1.43*eV, ... 

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 Brackett series (n=4, m>=5) in the infrared:

 frequencies (approximate):
 
 74*THz, 114*THz, 138*THz, 154*THz, 165*THz, 173*THz, 178*THz, 183*THz, 186*THz, ...

  wavelengths (4 digits, 10^3 nm = 1 micro (mu) m) : 

 4053*nm, 2627*nm, 2167*nm, 1946*nm, 1818*nm, 1737*nm, 1682*nm, 1642*nm, 1612*nm, ...

 energies (3 digits) meV (milli eV):

 306*meV, 472*meV, 572*meV, 638*meV, 682*meV, 714*meV, 738*meV, 756*meV, 769*meV, ...

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 Pfund series (n=5, m>=6) in the infrared:

 frequencies (approximate):
 
 40.2*THz, 64.5*THz, 80.2*THz, 91.0*THz, 98.7*THz, 104*THz, 109*THz, ...

 wavelengths (4 digits, 10^3 nm = 1 micro (mu) m) : 

 7462*nm, 4655*nm, 3742*nm, 3298*nm, 3040*nm, 2874*nm, 2759*nm, 2676*nm, ...

 energies (3 digits) meV (milli eV):

 166*meV, 266*meV, 332*meV, 376*meV, 408*meV, 432*meV, 450*meV, 463*meV, ...
 

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