A137525 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).

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

Offset

1,1

Comments

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}

Links

Table of n, a(n) for n=1..88.

Formula

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)).

Example

{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}

Mathematica

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]

Crossrefs

Sequence in context: A078924 A229141 A137526 * A166335 A109187 A265089

Adjacent sequences: A137522 A137523 A137524 * A137526 A137527 A137528

Keyword

uned,tabf,sign

Author

Roger L. Bagula, Apr 24 2008

Status

approved