%I A137981
%S A137981 2,3,30,92,120,696,720,8340,5220,24120,40296,103680,722160,289440,1216080,
%T A137981 756000,10579800,13003200,73306800,21281400,86350320,71284800,268531200,
%U A137981 2283140160,1799884800,9170280000,2072407680,8319024000,2438553600,41653241280
%V A137981 2,-3,30,-92,-120,696,720,-8340,-5220,24120,40296,103680,-722160,-289440,
               1216080,
%W A137981 -756000,10579800,13003200,-73306800,-21281400,86350320,-71284800,-268531200,
%X A137981 2283140160,1799884800,-9170280000,-2072407680,8319024000,2438553600,-41653241280
%N A137981 Triangular sequence of coefficients of an expansion of a tri-degenerate 
               partition of a Bernoulli B(x,n), Chebyshev U(x,n) and a Hermite H(x,
               n) Sheffer sequence: 1) b(x,t)=t*Exp(x*t)/(Exp(t)-1);; 2) u(x,t)=1/
               (1-2*x*t+t^2); 3) h(x,t)=Exp(2*x*t-t^2); p(x,t)=b(x,t)*u(x,t)*h(x,
               t).
%C A137981 Row sums are:
%C A137981 {2, 27, 484, 11280, 348456, 14589120, 819545280, 59538931200, 5399765562240, 
               596993441425920, 79100889171494400};
%C A137981 The atomic partition function ( Schodinger wave mechanic solution) is 
               ( of the type):
%C A137981 P(x,y,z,t)=Bessel(x,n)*Fourier(y,m)*Legendre(z,l)*Spin(t,k)
%C A137981 That differential equation solution is basic quantum chemistry.
%C A137981 This partition is a projection of an {x,y,z} solution to a simple {x}.
%C A137981 That is why I call it "degenerate".
%C A137981 In the study of blackbody radiation "actual observation" there is a longer 
               tail than the Bernoulli
%C A137981 distribution would predict ( actually a different curve shape).
%C A137981 That is what actually gave me this sort of idea.
%C A137981 Chebyshev is kind of special relativity velocity like with weight of 
               1/Sqrt[1-x^2].
%C A137981 Hermite is vibrator like/ oscillator with weight of Exp[ -x^2].
%C A137981 Bernoulli is "boson" like/ radiation and is used for most blackbody calculations
%C A137981 "classically". If you take the Hermite -Chebyshev as being a {3,1} type 
               special relativity like system,
%C A137981 them the surface of a molecule behaving in a relativistic fashion would 
               emit
%C A137981 photons by a partition like this...
%C A137981 This kind of thinking is what is called a physical "model" of a system.
%F A137981 1) b(x,t)=t*Exp(x*t)/(Exp(t)-1) 2) u(x,t)=1/(1-2*x*t+t^2) 3) 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)).
%e A137981 {2},
%e A137981 {-3, 30},
%e A137981 {-92, -120, 696},
%e A137981 {720, -8340, -5220, 24120},
%e A137981 {40296, 103680, -722160, -289440, 1216080},
%e A137981 {-756000, 10579800, 13003200, -73306800, -21281400, 86350320},
%e A137981 {-71284800, -268531200, 2283140160, 1799884800, -9170280000, -2072407680,
               8319024000},
%e A137981 {2438553600, -41653241280, -75214137600, 512773208640, 294226732800, 
               -1419973571520, -262049256000, 1048990642560},
%e A137981 {363233082240, 1780144128000, -18252909964800, -21270283776000, 128466991699200, 
               57669354086400, -269259449356800, -41959625702400, 167862311366400}, 
               {-19615115520000, 402989255222400, 953596385280000, -7683295115865600, 
               -6554013637248000, 36694836029617920, 13501295252121600, -61627138532467200, 
               -8309184412636800, 33237523332921600},
%e A137981 {-4323226030502400, -25964377528320000, 313773569755430400, 482205929103360000, 
               -3370903650930816000, -2258955163745280000, 12006293764671360000, 
               3739168341393408000, -16784976782008512000, -1994251399975296000, 
               7977033884466662400}
%t A137981 Clear[p, b, a]; 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]
%Y A137981 Sequence in context: A024631 A032814 A095927 this_sequence A110351 A088115 
               A048986
%Y A137981 Adjacent sequences: A137978 A137979 A137980 this_sequence A137982 A137983 
               A137984
%K A137981 tabl,uned,sign
%O A137981 1,1
%A A137981 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), May 01 2008