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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 Table of n, a(n) for n=1..53. 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 Adjacent sequences: A138021 A138022 A138023 * A138025 A138026 A138027 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.)