login
Triangular sequence of coefficients of p(x,t) = t*exp(3*x*t - t^2)/(exp(t) - 1).
0

%I #18 Jan 17 2022 16:49:09

%S 2,-3,18,-44,-72,216,360,-1980,-1620,3240,7176,25920,-71280,-38880,

%T 58320,-151200,753480,1360800,-2494800,-1020600,1224720,-3587520,

%U -21772800,54250560,65318400,-89812800,-29393280,29393280,152409600,-678041280,-2057529600,3417785280,3086294400

%N Triangular sequence of coefficients of p(x,t) = t*exp(3*x*t - t^2)/(exp(t) - 1).

%D Steve Roman, The Umbral Calculus, Dover Publications, New York (1984), pp. 87-100.

%D Frederick T. Wall, Chemical Thermodynamics, W. H. Freeman, San Francisco (1965), pp. 282-290.

%e Triangle begins

%e 2;

%e -3, 18;

%e -44, -72, 216;

%e 360, -1980, -1620, 3240;

%e 7176, 25920, -71280, -38880, 58320;

%e -151200, 753480, 1360800, -2494800, -1020600, 1224720;

%e -3587520, -21772800, 54250560, 65318400, -89812800, -29393280, 29393280;

%e ...

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

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

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

%t Flatten[a]

%K tabl,sign

%O 1,1

%A _Roger L. Bagula_, Apr 28 2008