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A120072
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Numerator triangle for hydrogen spectrum rationals.
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17
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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
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OFFSET
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2,1
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COMMENTS
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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):= A120072(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.
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LINKS
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Table of n, a(n) for n=2..60.
W. 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
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FORMULA
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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):=A120072(m,n)/A120073(m,n), m>=2, 1<= n<=m-1, are given in the W. Lang link.
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MATHEMATICA
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Table[1/n^2 - 1/m^2, {m, 2, 12}, {n, 1, m-1}] // Flatten // Numerator (* Jean-François Alcover, Sep 16 2013 *)
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CROSSREFS
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Row sums give A120074. Row sums of r(m, n) triangle give A120076(m)/A120077(m), m>=2.
Cf. A126252.
Sequence in context: A229598 A078356 A050093 * A166492 A120070 A143753
Adjacent sequences: A120069 A120070 A120071 * A120073 A120074 A120075
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KEYWORD
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nonn,easy,tabl,frac
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AUTHOR
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Wolfdieter Lang, Jul 20 2006
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STATUS
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approved
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