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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

%I #11 Dec 09 2017 19:39:00

%S 1,-1,1,2,-4,2,-6,12,-12,4,24,-48,48,-32,8,-120,240,-240,160,-80,16,

%T 720,-1440,1440,-960,480,-192,32,-5040,10080,-10080,6720,-3360,1344,

%U -448,64,40320,-80640,80640,-53760,26880,-10752,3584,-1024,128,-362880,725760,-725760,483840,-241920,96768,-32256,9216

%N 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).

%C Row sums are {1, 0, 0, -2, 0, -24, 80, -720, 5376, -49280, 490752}.

%D A. H. W. Beck, Space-Charge Waves and Slow Electromagnetic Waves, Pergamon Press, New York, 1958, page 30

%D A. M. Kuethe, J. D. Schetzer, Foundations of Aerodynamics, John Wiley and sons, Inc, New York, page 177

%F 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)).

%e {1},

%e {-1, 1},

%e {2, -4, 2},

%e {-6, 12, -12, 4},

%e {24, -48, 48, -32, 8},

%e {-120, 240, -240, 160, -80, 16},

%e {720, -1440, 1440, -960, 480, -192, 32},

%e {-5040, 10080, -10080, 6720, -3360, 1344, -448, 64},

%e {40320, -80640, 80640, -53760, 26880, -10752, 3584, -1024, 128},

%e {-362880, 725760, -725760, 483840, -241920, 96768, -32256, 9216, -2304, 256}, {3628800, -7257600, 7257600, -4838400, 2419200, -967680, 322560, -92160, 23040, -5120, 512}

%t p[t_] = FullSimplify[1 - Exp[2*x*t]*(t - 1)/(1 + t)];

%t g = Table[ ExpandAll[(n!/2)*SeriesCoefficient[Series[p[t], {t, 0, 30}], n]], {n, 0, 10}];

%t a = Table[ CoefficientList[(n!/2)*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n], x], {n, 0, 10}]; Flatten[a]

%K uned,tabl,sign

%O 1,4

%A _Roger L. Bagula_, May 01 2008

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