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A120072 Numerator triangle for hydrogen spectrum rationals.

%I A120072 #34 Apr 25 2023 20:18:33
%S A120072 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 A120072 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 A120072 85,72,57,40,21,143,35,5,1,119,1,95,5,7,11,23
%N A120072 Numerator triangle for hydrogen spectrum rationals.
%C A120072 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 A120072 The spectral series for n=1,2,...,7, m>=n+1, are named after Lyman, Balmer, Paschen, Brackett, Pfund, Humphreys, Hansen-Strong, respectively.
%C A120072 The corresponding denominator triangle is A120073.
%C A120072 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 A120072 G. C. Greubel, <a href="/A120072/b120072.txt">Rows n = 2..50 of the triangle, flattened</a>
%H A120072 Wolfdieter Lang, <a href="/A120072/a120072.txt">First ten rows, rationals and more</a>.
%H A120072 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 A120072 a(m,n) = numerator(r(m,n)) with r(m,n) = 1/n^2 - 1/m^2, m>=2, n=1..m-1.
%F A120072 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 A120072 For the rational triangle see W. Lang link.
%e A120072 Numerator triangle begins as:
%e A120072    3;
%e A120072    8,  5;
%e A120072   15,  3,  7;
%e A120072   24, 21, 16,  9;
%e A120072   35,  2,  1,  5, 11;
%e A120072   48, 45, 40, 33, 24, 13;
%e A120072   63, 15, 55,  3, 39,  7, 15;
%e A120072   80, 77,  8, 65, 56,  5, 32, 17;
%e A120072   99,  6, 91, 21,  3,  4, 51,  9, 19;
%t A120072 Table[1/n^2 - 1/m^2, {m,2,12}, {n,m-1}]//Flatten//Numerator (* _Jean-François Alcover_, Sep 16 2013 *)
%o A120072 (Magma) [Numerator(1/k^2 - 1/n^2): k in [1..n-1], n in [2..18]]; // _G. C. Greubel_, Apr 24 2023
%o A120072 (SageMath)
%o A120072 def A120072(n,k): return numerator(1/k^2 - 1/n^2)
%o A120072 flatten([[A120072(n,k) for k in range(1,n)] for n in range(2,19)]) # _G. C. Greubel_, Apr 24 2023
%Y A120072 Row sums give A120074.
%Y A120072 Row sums of r(m, n) triangle give A120076(m)/A120077(m), m>=2.
%Y A120072 Cf. A120070, A120073, A120075, A126252.
%K A120072 nonn,easy,tabl,frac
%O A120072 2,1
%A A120072 _Wolfdieter Lang_, Jul 20 2006