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Search: id:A138022
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| A138022 |
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Triangular sequence of an expansion of a receding Doppler frequency as a Sheffer sequence: nu'=nu*Sqrt[1-v^2/c^2]/(1+v/c); Substituting: nu->Exp[t*x];v/c->t; to give: p(x,t)=Sqrt[1 - t^2]*Exp[x*t]/(1 + t). |
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+0
1
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| 1, -1, 1, 1, -2, 1, -3, 3, -3, 1, 9, -12, 6, -4, 1, -45, 45, -30, 10, -5, 1, 225, -270, 135, -60, 15, -6, 1, -1575, 1575, -945, 315, -105, 21, -7, 1, 11025, -12600, 6300, -2520, 630, -168, 28, -8, 1, -99225, 99225, -56700, 18900, -5670, 1134, -252, 36, -9, 1, 893025, -992250, 496125, -189000, 47250, -11340
(list; table; graph; listen)
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OFFSET
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1,5
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COMMENT
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Row sums are:A130915;
{1, 0, 0, -2, 0, -24, 40, -720, 2688, -42560, 245376};
This sequence is a thought experiment in what if the expansion red shift numbers
we observe involve an interaction of light with the vacuum in a quantum manner.
( the exp[x*t] as an U(1) interaction of the vacuum with the frequency).
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REFERENCES
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Lloyd Motz, Anneta Duveen, Essential of Astronomy, Wadsworth Publishing, Belmont, Ca., 1967, page 176
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FORMULA
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p(x,t)=Sqrt[1 - t^2]*Exp[x*t]/(1 + t)=sum(P(x,n)*t^n/n!,{n,0,Infinity}); out_n,m=n!*Coefficients(P(x,n)).
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EXAMPLE
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{1},
{-1, 1},
{1, -2, 1},
{-3, 3, -3, 1},
{9, -12, 6, -4, 1},
{-45, 45, -30, 10, -5, 1},
{225, -270, 135, -60, 15, -6, 1},
{-1575, 1575, -945,315, -105, 21, -7, 1},
{11025, -12600, 6300, -2520, 630, -168, 28, -8, 1},
{-99225, 99225, -56700, 18900, -5670, 1134, -252, 36, -9, 1},
{893025, -992250, 496125, -189000, 47250, -11340, 1890, -360, 45, -10, 1}
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MATHEMATICA
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Clear[p, b, a]; p[t_] = FullSimplify[Sqrt[1 - t^2]*Exp[x*t]/(1 + t)]; g = Table[ ExpandAll[n!*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]], {n, 0, 10}]; a = Table[ CoefficientList[n!*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n], x], {n, 0, 10}]; Flatten[a]
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CROSSREFS
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Cf. A130915.
Sequence in context: A039775 A152534 A136018 this_sequence A113278 A132382 A048865
Adjacent sequences: A138019 A138020 A138021 this_sequence A138023 A138024 A138025
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KEYWORD
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tabl,uned,sign
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AUTHOR
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Roger L. Bagula (rlbagulatftn(AT)yahoo.com), May 01 2008
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