a(m,n) tabl head (triangle) for  A120070 (used for frequencies, energies, wave lengths of H-Atom spectrum)
 

    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     12      7      0     0     0     0     0     0     0

    5       24     21     16      9     0     0     0     0     0     0

    6       35     32     27     20    11     0     0     0     0     0

    7       48     45     40     33    24    13     0     0     0     0

    8       63     60     55     48    39    28    15     0     0     0

    9       80     77     72     65    56    45    32    17     0     0

   10       99     96     91     84    75    64    51    36    19     0

   11      120    117    112    105    96    85    72    57    40    21
 
   .
   .
   .

 The generating functon for the column n numbers is A(n,x):=sum(a(m,n)*x^m,m=1..n-1) = 

 (x^(n+1))*((2*n+1)- (2*n-1)*x)/(1-x)^3.
 
 Remark: 

 In the calculation of these o.g.f.s one encounters the row polynomials of triangle A094728:
 
 G(n,x):= sum(a(m,n)*x^m,m=n+1..infty) =  sum(a(m,n)*x^m,m=0..infty) - sum(a(m,n)*x^m,m=0..n)

        = (x d_x)^2 (1/(1-x)) - (n^2)/(1-x) + T(n,x), with T(n,x) the polynomial of row n of 

  triangle A094728(n,k).
  
  E.g.: n=3: G(3,x) = (19*x-8*x^2-9)/(1-x)^3 + T(3,x) =  (19*x-8*x^2-9)/(1-x)^3 + (9 + 8*x +5*x^2) = 

  (x^4)*(7-5*x)/(1-x)^3.

 ###################################################################################################

 The generating function for the row sums  [3, 13, 34, 70, 125, 203, 308, 444, 615, 825, ...]  = 

 A016061(n-1),n>=2, is x^2*(3+x)/(1-x)^4.


 ###################################################################################################################

 The rationals r(m,n):= a(m,n)/(m^2*n^2) are found under  A120072 (numerators) and A120073 (denominators): 

 See also the W. Lang link under  A120072 for the r(m,n) table and the column o.g.f.s.

 There also the frequencies, energies and wave lengths of the H-spectrum series for n=1 (Lyman), n=2 (Balmer),

 n=3 (Paschen), n=4 (Brackett) and n=5 (Pfund) series are given.
 

############################################### e.o.f.##############################################################