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 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). 0

%I #12 Jul 26 2021 01:17:25

%S 2,-3,30,-92,-120,696,720,-8340,-5220,24120,40296,103680,-722160,

%T -289440,1216080,-756000,10579800,13003200,-73306800,-21281400,

%U 86350320,-71284800,-268531200,2283140160,1799884800,-9170280000,-2072407680,8319024000,2438553600,-41653241280

%N 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).

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

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

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

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

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

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

%e {2},

%e {-3, 30},

%e {-92, -120, 696},

%e {720, -8340, -5220, 24120},

%e {40296, 103680, -722160, -289440, 1216080},

%e {-756000, 10579800, 13003200, -73306800, -21281400, 86350320},

%e {-71284800, -268531200, 2283140160, 1799884800, -9170280000, -2072407680, 8319024000},

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

%O 1,1

%A _Roger L. Bagula_, May 01 2008

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Last modified May 29 03:48 EDT 2024. Contains 372921 sequences. (Running on oeis4.)