A120072 Numerator triangle for hydrogen spectrum rationals.

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

2,1

Comments

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.

Links

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

Formula

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.

Example

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;

Mathematica

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

Prog

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

Crossrefs

Row sums give A120074.

Row sums of r(m, n) triangle give A120076(m)/A120077(m), m>=2.

Cf. A120070, A120073, A120075, A126252.

Sequence in context: A229598 A078356 A050093 * A166492 A120070 A143753

Adjacent sequences: A120069 A120070 A120071 * A120073 A120074 A120075

Keyword

nonn,easy,tabl,frac

Author

Wolfdieter Lang, Jul 20 2006

Status

approved