%I #3 Mar 30 2012 17:34:26
%S -8,0,-48,0,64,0,-960,0,3072,0,-1536,-40320,0,207360,0,-230400,0,
%T 61440,0,-2903040,0,20643840,0,-36126720,0,20643840,0,-3440640,0,
%U -319334400,0,2903040000,0,-6967296000,0,6193152000,0,-2167603200,0,247726080
%N A triangular sequence of coefficients based on an expansion of a Skew Catenoid of Matthius Weber like Sheffer expansion function: g(t) = x*(3 + t^2)/(t^2 - 1); f(t) = (3 + t^2)/(t^2 - 1); p(x,t) = Exp[x*(t)]*(1 - f(t)2).
%C Row sums:
%C {-8, 0, 16, 0, 576, 0, -1920, 0, -1182720, 0, -110315520}
%C Here is the relationship that seems to hold:
%C Weierstrass{f,g)-> Sheffer{g,fbar}.
%D Bloomington's Virtual Minimal Surface Museum; Matthias Weber,http://www.indiana.edu/~minimal/toc.html Lecture 2.
%F g(t) = x*(3 + t^2)/(t^2 - 1); f(t) = (3 + t^2)/(t^2 - 1); p(x,t) = Exp[x*(t)]*(1 - f(t)2)=Sum(P(x,n)*t^n/n!,{n,0,Infinity}]; out_n,m=n!*Coefficients(P(x,n)/Exp[ -3*x^2).
%e {-8},
%e {0},
%e {-48, 0, 64},
%e {0},
%e {-960, 0, 3072, 0, -1536},
%e {},
%e {-40320, 0, 207360, 0, -230400, 0, 61440},
%e {0},
%e {-2903040, 0,20643840, 0, -36126720, 0, 20643840, 0, -3440640},
%e {0}, {-319334400, 0, 2903040000, 0, -6967296000, 0, 6193152000, 0, -2167603200, 0, 247726080}
%t Clear[p, f, g] g[t_] = x*(3 + t^2)/(t^2 - 1); f[t] = (3 + t^2)/(t^2 - 1); p[t_] = Exp[x*g[t]]*(1 - f[t]^2); g = Table[ FullSimplify[ExpandAll[(n!)*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]]/Exp[ -3*x^2]], {n, 0, 10}]; a = Table[ CoefficientList[n!*SeriesCoefficient[ FullSimplify[Series[p[t], {t, 0, 30}]], n]/Exp[ -3*x^2], x], {n, 0, 10}]; Flatten[a]
%K uned,tabl,sign
%O 1,1
%A _Roger L. Bagula_, Apr 24 2008
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