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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 August 23 11:24 EDT 2019. Contains 326222 sequences. (Running on oeis4.)