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A137525
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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).
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0
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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, 0, 5040, 0, -120, 0, 0, 0, 0, 0, 0, 40320, 0, -720, 0, 0, 0, 0, 0, 0, 0, 362880, 0, -5040, 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
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OFFSET
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1,1
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COMMENTS
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Row sums:
{1, 5, 22, 114, 696, 4920, 39600, 357840, 3588480, 39553920, 475372800};
Since this sequence seems to start at the quadratic level,
there may be lower extensions to it.
When I started doing Sheffer sequences I noticed the similarity between the
{f,g} data of Weierstrass definitions of minimal surfaces and the two function
Sheffer sequence generators.
Here is the relationship that seems to hold:
Weierstrass{f,g)-> Sheffer{g,fbar}
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LINKS
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FORMULA
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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)).
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EXAMPLE
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{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, 0, 5040, 0, -120},
{0, 0, 0, 0, 0, 0, 40320, 0, -720},
{0, 0, 0, 0, 0, 0, 0, 362880, 0, -5040},
{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}
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MATHEMATICA
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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]
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CROSSREFS
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KEYWORD
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uned,tabf,sign
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AUTHOR
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STATUS
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approved
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