%I A137778
%S A137778 1,2,2,4,8,4,12,36,36,12,48,192,288,192,48,240,1200,2400,2400,1200,240,
%T A137778 1440,8640,21600,28800,21600,8640,1440,10080,70560,211680,352800,352800,
%U A137778 211680,70560,10080,80640,645120,2257920,4515840,5644800,4515840
%V A137778 -1,-2,-2,-4,-8,-4,-12,-36,-36,-12,-48,-192,-288,-192,-48,-240,-1200,-2400,
               -2400,-1200,
%W A137778 -240,-1440,-8640,-21600,-28800,-21600,-8640,-1440,-10080,-70560,-211680,
               -352800,
%X A137778 -352800,-211680,-70560,-10080,-80640,-645120,-2257920,-4515840,-5644800,
               -4515840
%N 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)).
%C A137778 Row sums = A032184.
%C A137778 But for an odd function of 2^n this might be a simple:
%C A137778 p[x,n]= -f(2^n)*((x+1)/(x-1))^n/n!;
%C A137778 The importance of these density curves is
%C A137778 that they are related to Mach's numbers for
%C A137778 velocity in a medium.
%C A137778 It also seems important that the equation has a Moebius form that is
%C A137778 Blaschke/ Elliptic in shape in terms of gamma and pressure ratio pr:
%C A137778 F(pr)=(f(gamma)+pr)/(1+f(gamma)*pr).
%D A137778 A. M. Kuethe, J.D. Schetzer,Foundations of Aerodynamics, John Wiley and 
               sons, Inc. New York,1959, page 180
%F A137778 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).
%e A137778 {-1},
%e A137778 {-2, -2},
%e A137778 {-4, -8, -4},
%e A137778 {-12, -36, -36, -12},
%e A137778 {-48, -192, -288, -192, -48},
%e A137778 {-240, -1200, -2400, -2400, -1200, -240},
%e A137778 {-1440, -8640, -21600, -28800, -21600, -8640, -1440},
%e A137778 {-10080, -70560, -211680, -352800, -352800, -211680, -70560, -10080},
%e A137778 {-80640, -645120, -2257920, -4515840, -5644800, -4515840, -2257920, -645120, 
               -80640},
%e A137778 {-725760, -6531840, -26127360, -60963840, -91445760, -91445760, -60963840, 
               -26127360, -6531840, -725760},
%e A137778 {-7257600, -72576000, -326592000, -870912000, -1524096000, -1828915200, 
               -1524096000, -870912000, -326592000, -72576000, -7257600}
%t A137778 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}];
%Y A137778 Cf. A007318, A032184.
%Y A137778 Sequence in context: A121175 A038208 A116694 this_sequence A000017 A032522 
               A077964
%Y A137778 Adjacent sequences: A137775 A137776 A137777 this_sequence A137779 A137780 
               A137781
%K A137778 tabl,uned,sign
%O A137778 1,2
%A A137778 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Apr 28 2008