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Numerator triangle for hydrogen spectrum rationals.
19

%I #34 Apr 25 2023 20:18:33

%S 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,

%T 7,15,80,77,8,65,56,5,32,17,99,6,91,21,3,4,51,9,19,120,117,112,105,96,

%U 85,72,57,40,21,143,35,5,1,119,1,95,5,7,11,23

%N Numerator triangle for hydrogen spectrum rationals.

%C 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.

%C The spectral series for n=1,2,...,7, m>=n+1, are named after Lyman, Balmer, Paschen, Brackett, Pfund, Humphreys, Hansen-Strong, respectively.

%C The corresponding denominator triangle is A120073.

%C 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.

%H G. C. Greubel, <a href="/A120072/b120072.txt">Rows n = 2..50 of the triangle, flattened</a>

%H Wolfdieter Lang, <a href="/A120072/a120072.txt">First ten rows, rationals and more</a>.

%H T. Lyman, <a href="http://adsabs.harvard.edu/full/1906ApJ....23..181L">The Spectrum of Hydrogen in the Region of Extremely Short Wave-Lengths</a>, The Astrophysical Journal, 23 (April 1906), 181-210. - _Paul Curtz_, May 30 2017

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

%F 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.

%e For the rational triangle see W. Lang link.

%e Numerator triangle begins as:

%e 3;

%e 8, 5;

%e 15, 3, 7;

%e 24, 21, 16, 9;

%e 35, 2, 1, 5, 11;

%e 48, 45, 40, 33, 24, 13;

%e 63, 15, 55, 3, 39, 7, 15;

%e 80, 77, 8, 65, 56, 5, 32, 17;

%e 99, 6, 91, 21, 3, 4, 51, 9, 19;

%t Table[1/n^2 - 1/m^2, {m,2,12}, {n,m-1}]//Flatten//Numerator (* _Jean-François Alcover_, Sep 16 2013 *)

%o (Magma) [Numerator(1/k^2 - 1/n^2): k in [1..n-1], n in [2..18]]; // _G. C. Greubel_, Apr 24 2023

%o (SageMath)

%o def A120072(n,k): return numerator(1/k^2 - 1/n^2)

%o flatten([[A120072(n,k) for k in range(1,n)] for n in range(2,19)]) # _G. C. Greubel_, Apr 24 2023

%Y Row sums give A120074.

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

%Y Cf. A120070, A120073, A120075, A126252.

%K nonn,easy,tabl,frac

%O 2,1

%A _Wolfdieter Lang_, Jul 20 2006