A137778 Triangular sequence from coefficients of an expansion of a Rankine-Hugoniot relation function for density in terms of thermodynamic gamma as t and pressure ratio as x: p(x,t)=((t + 1)/(t - 1) + x)/(1 + (t + 1)*x/(t - 1)).
-1, -2, -2, -4, -8, -4, -12, -36, -36, -12, -48, -192, -288, -192, -48, -240, -1200, -2400, -2400, -1200, -240, -1440, -8640, -21600, -28800, -21600, -8640, -1440, -10080, -70560, -211680, -352800, -352800, -211680, -70560, -10080, -80640, -645120, -2257920, -4515840, -5644800, -4515840
Offset: 1
Examples
{-1},
{-2, -2},
{-4, -8, -4},
{-12, -36, -36, -12},
{-48, -192, -288, -192, -48},
{-240, -1200, -2400, -2400, -1200, -240},
{-1440, -8640, -21600, -28800, -21600, -8640, -1440},
{-10080, -70560, -211680, -352800, -352800, -211680, -70560, -10080},
{-80640, -645120, -2257920, -4515840, -5644800, -4515840, -2257920, -645120, -80640},
{-725760, -6531840, -26127360, -60963840, -91445760, -91445760, -60963840, -26127360, -6531840, -725760},
{-7257600, -72576000, -326592000, -870912000, -1524096000, -1828915200, -1524096000, -870912000, -326592000, -72576000, -7257600}
References
- A. M. Kuethe, J.D. Schetzer, Foundations of Aerodynamics, John Wiley and sons, Inc. New York,1959, page 180
Programs
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Mathematica
p[t_] = ((t + 1)/(t - 1) + x)/(1 + (t + 1)*x/(t - 1)); Table[ExpandAll[ FullSimplify[(n!*(1 - x)^(n))*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]]], {n, 0, 10}]; a = Table[ CoefficientList[ExpandAll[ FullSimplify[(n!*(1 - x)^(n))*SeriesCoefficient[Series[p[t], {t, 0, 30}], n]]], x], {n, 0, 10}]; Flatten[a] Table[ Apply[Plus, CoefficientList[ExpandAll[ FullSimplify[(n!*(1 - x)^(n))*SeriesCoefficient[Series[p[t], {t, 0, 30}], n]]], x]], {n, 0, 10}];
Formula
p(x,t)=((t + 1)/(t - 1) + x)/(1 + (t + 1)*x/(t -1))=Sum(Q(x,n)*t^n/n!,{n,0,Infinity}]; out_n,m]=n!*(1 - x)^(n))*Coefficient(Q(x,n).
Comments