A137981 Triangle read by rows: expansion of p(x,t) = b(x,t)*u(x,t)*h(x,t) where b(x,t) = t*exp(x*t)/(exp(t)-1), u(x,t) = 1/(1-2*x*t+t^2), and h(x,t) = exp(2*x*t-t^2).

2, -3, 30, -92, -120, 696, 720, -8340, -5220, 24120, 40296, 103680, -722160, -289440, 1216080, -756000, 10579800, 13003200, -73306800, -21281400, 86350320, -71284800, -268531200, 2283140160, 1799884800, -9170280000, -2072407680, 8319024000, 2438553600, -41653241280

Offset

1,1

Comments

This kind of thinking is what is called a physical "model" of a system.

Links

Table of n, a(n) for n=1..30.

Formula

b(x,t)=t*Exp(x*t)/(Exp(t)-1)

u(x,t)=1/(1-2*x*t+t^2)

h(x,t)=Exp(2*x*t-t^2)

p(x,t)=b(x,t)*u(x,t)*h(x,t)=sum(P(x,n)*t^n/n!,{n,0,Infinity});

out_n,m=(n + 2)!*n!*Coefficients(P(x,n)).

Example

{2},
{-3, 30},
{-92, -120, 696},
{720, -8340, -5220, 24120},
{40296, 103680, -722160, -289440, 1216080},
{-756000, 10579800, 13003200, -73306800, -21281400, 86350320},
{-71284800, -268531200, 2283140160, 1799884800, -9170280000, -2072407680, 8319024000},

Mathematica

p[t_] = FullSimplify[(t*Exp[x*t]/(Exp[t] - 1))*(Exp[2*x*t - t^2])/(1 - 2*x*t + t^2)];
Table[ ExpandAll[(n + 2)!*n!*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]], {n, 0, 10}];
a = Table[ CoefficientList[(n + 2)!*n!*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n], x], {n, 0, 10}];
Flatten[a]

Crossrefs

Sequence in context: A296248 A325506 A203431 * A110351 A326224 A270483

Adjacent sequences: A137978 A137979 A137980 * A137982 A137983 A137984

Keyword

tabl,uned,sign

Author

Roger L. Bagula, May 01 2008

Status

approved