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A138024 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). 0
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; refs; listen; history; text; internal format)
OFFSET
1,4
COMMENTS
Row sums are {1, 0, 0, -2, 0, -24, 80, -720, 5376, -49280, 490752}.
REFERENCES
A. H. W. Beck, Space-Charge Waves and Slow Electromagnetic Waves, Pergamon Press, New York, 1958, page 30
A. M. Kuethe, J. D. Schetzer, Foundations of Aerodynamics, John Wiley and sons, Inc, New York, page 177
LINKS
FORMULA
p(x,t)=1 - exp(2*x*t)*(t - 1)/(1 + t) = Sum_{n>=0} (P(x,n)*t^n/n!); out_n,m = (n!/2)*Coefficients(P(x,n)).
EXAMPLE
{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}
MATHEMATICA
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]
CROSSREFS
Sequence in context: A013599 A276528 A205839 * A167656 A253666 A174298
KEYWORD
uned,tabl,sign
AUTHOR
Roger L. Bagula, May 01 2008
STATUS
approved

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Last modified April 13 12:47 EDT 2024. Contains 371641 sequences. (Running on oeis4.)