A137528 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).

-8, 0, -48, 0, 64, 0, -960, 0, 3072, 0, -1536, -40320, 0, 207360, 0, -230400, 0, 61440, 0, -2903040, 0, 20643840, 0, -36126720, 0, 20643840, 0, -3440640, 0, -319334400, 0, 2903040000, 0, -6967296000, 0, 6193152000, 0, -2167603200, 0, 247726080

Offset

1,1

Comments

Row sums:

{-8, 0, 16, 0, 576, 0, -1920, 0, -1182720, 0, -110315520}

Here is the relationship that seems to hold:

Weierstrass{f,g)-> Sheffer{g,fbar}.

References

Bloomington's Virtual Minimal Surface Museum; Matthias Weber,http://www.indiana.edu/~minimal/toc.html Lecture 2.

Links

Table of n, a(n) for n=1..40.

Formula

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).

Example

{-8},
{0},
{-48, 0, 64},
{0},
{-960, 0, 3072, 0, -1536},
{},
{-40320, 0, 207360, 0, -230400, 0, 61440},
{0},
{-2903040, 0,20643840, 0, -36126720, 0, 20643840, 0, -3440640},
{0}, {-319334400, 0, 2903040000, 0, -6967296000, 0, 6193152000, 0, -2167603200, 0, 247726080}

Mathematica

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]

Crossrefs

Sequence in context: A340953 A047771 A298099 * A270006 A167318 A186979

Adjacent sequences: A137525 A137526 A137527 * A137529 A137530 A137531

Keyword

uned,tabl,sign

Author

Roger L. Bagula, Apr 24 2008

Status

approved