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Search: id:A138024
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| A138024 |
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A triangular sequence of coefficients of an expansion of a Mach wave as a traveling wave in a medium: (vt')^2=vp*vg=c^2-(gamma-1)/gamma+1)*vt^2; Substituting: vt-> Exp[t*x];gamma->t;c->1; p(x,t)=1 - Exp[2*x*t]*(t - 1)/(1 + t). |
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+0
1
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| 1, -1, 1, 2, -4, 2, -6, 12, -12, 4, 24, -48, 48, -32, 8, -120, 240, -240, 160, -80, 16, 720, -1440, 1440, -960, 480, -192, 32, -5040, 10080, -10080, 6720, -3360, 1344, -448, 64, 40320, -80640, 80640, -53760, 26880, -10752, 3584, -1024, 128, -362880, 725760, -725760, 483840, -241920, 96768, -32256, 9216
(list; table; graph; listen)
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OFFSET
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1,4
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COMMENT
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Row sums are: {1, 0, 0, -2, 0, -24, 80, -720, 5376, -49280, 490752};
The idea here is that gamma is very close to one in ordinary terms
so that the usually measured:
vt=c=Sqrt[vp*vg];
but that over long distances an interaction takes place.
The Doppler model gives row sums of:
{1, 0, 0, -2, 0, -24, 40, -720, 2688, -42560, 245376}
which are very roughly of the same order.
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REFERENCES
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A.H,. W. Beck, Space -Charge Waves and Slow Electromagnetic Waves,Pergamon Press, New York, 1958, page 30
A. M Kuete, J. D. Schetzer, Foundations of Areodynamnics, John Wiley and sons, Inc, New York, page 177
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FORMULA
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p(x,t)=1 - Exp[2*x*t]*(t - 1)/(1 + t)=Sum(P(x,n)*t^n/n!,{n,0,Infinity}); out_n,m=(n!/2)*Coefficients(P(x,n)).
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EXAMPLE
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{1},
{-1, 1},
{2, -4, 2},
{-6, 12, -12, 4},
{24, -48, 48, -32, 8},
{-120, 240, -240, 160, -80, 16},
{720, -1440, 1440, -960, 480, -192, 32},
{-5040, 10080, -10080, 6720, -3360, 1344, -448, 64},
{40320, -80640, 80640, -53760, 26880, -10752, 3584, -1024, 128},
{-362880, 725760, -725760, 483840, -241920, 96768, -32256, 9216, -2304, 256}, {3628800, -7257600, 7257600, -4838400, 2419200, -967680, 322560, -92160, 23040, -5120, 512}
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MATHEMATICA
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Clear[p, b, a]; p[t_] = FullSimplify[1 - Exp[2*x*t]*(t - 1)/(1 + t)]; g = Table[ ExpandAll[(n!/2)*SeriesCoefficient[Series[p[t], {t, 0, 30}], n]], {n, 0, 10}]; a = Table[ CoefficientList[(n!/2)*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n], x], {n, 0, 10}]; Flatten[a]
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CROSSREFS
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Adjacent sequences: A138021 A138022 A138023 this_sequence A138025 A138026 A138027
Sequence in context: A085190 A104000 A013599 this_sequence A021012 A021416 A094756
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KEYWORD
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uned,tabl,sign
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AUTHOR
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Roger L. Bagula (rlbagulatftn(AT)yahoo.com), May 01 2008
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