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%I #3 Mar 30 2012 17:34:26
%S 2,0,-1,0,6,0,-1,0,0,24,0,-2,0,0,0,120,0,-6,0,0,0,0,720,0,-24,0,0,0,0,
%T 0,5040,0,-120,0,0,0,0,0,0,40320,0,-720,0,0,0,0,0,0,0,362880,0,-5040,
%U 0,0,0,0,0,0,0,0,3628800,0,-40320,0,0,0,0,0,0,0,0,0,39916800,0,-362880,0,0,0,0,0,0,0,0,0,0,479001600,0,-3628800
%N A triangular sequence of coefficients based on an expansion of a Catenoid like Sheffer expansion function: g(t) = t; f(t) = -1/t; p(x,t) = Exp[x*(t)]*(1 - f(t)^2).
%C Row sums:
%C {1, 5, 22, 114, 696, 4920, 39600, 357840, 3588480, 39553920, 475372800};
%C Since this sequence seems to start at the quadratic level,
%C there may be lower extensions to it.
%C When I started doing Sheffer sequences I noticed the similarity between the
%C {f,g} data of Weierstrass definitions of minimal surfaces and the two function
%C Sheffer sequence generators.
%C Here is the relationship that seems to hold:
%C Weierstrass{f,g)-> Sheffer{g,fbar}
%F g(t) = t; f(t) = -1/t; p(x,t) = Exp[x*(t)]*(1 - f(t)^2)=Sum(P(x,n)*t^n/n!,{n,0,Infinity}]; out_n,m=n!*(n+2)!*Coefficients(P(x,n)).
%e {2, 0, -1},
%e {0, 6, 0, -1},
%e {0, 0, 24, 0, -2},
%e {0, 0, 0, 120, 0, -6},
%e {0, 0, 0, 0, 720, 0, -24},
%e {0, 0, 0, 0, 0, 5040, 0, -120},
%e {0, 0, 0, 0, 0, 0, 40320, 0, -720},
%e {0, 0, 0, 0, 0, 0, 0, 362880, 0, -5040},
%e {0, 0, 0, 0, 0, 0, 0, 0, 3628800, 0, -40320},
%e {0, 0, 0, 0, 0, 0, 0, 0, 0, 39916800, 0, -362880},
%e {0, 0, 0, 0,0, 0, 0, 0, 0, 0, 479001600, 0, -3628800}
%t Clear[p, f, g] g[t_] = t; f[t] = -1/t; p[t_] = Exp[x*g[t]]*(1 - f[t]^2); g = Table[ FullSimplify[ExpandAll[(n!)*(n + 2)!*SeriesCoefficient[Series[p[t], {t, 0, 30}], n]]], {n, 0, 10}]; a = Table[ CoefficientList[n!*(n + 2)!*SeriesCoefficient[ FullSimplify[Series[p[t], {t, 0, 30}]], n], x], {n, 0, 10}]; Flatten[a]
%K uned,tabf,sign
%O 1,1
%A _Roger L. Bagula_, Apr 24 2008
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