A137522 A triangular sequence from an expansion of coefficients of the function: p(x,t)=Exp(x*g*(t))*(1-f(t)^2);f(t)=1/Sqrt[1 - 14*t^4 + t^8];g(t)=t. (Based on the Weierstrass functions of Schwarz's minimal surface which is identified with a cube.)

0, 0, 0, 0, -336, 0, -1680, 0, 0, -5040, 0, 0, 0, -11760, -7862400, 0, 0, 0, -23520, 0, -70761600, 0, 0, 0, -42336, 0, 0, -353808000, 0, 0, 0, -70560

Offset

1,5

Comments

Row sums: {0, 0, 0, 0, -336, -1680, -5040, -11760, -7885920, -70803936, -353878560}.

Because of the 8th power in generator function nothing shows up until n=5 and then the secondary polynomial doesn't show up until the 9th power.

Links

Table of n, a(n) for n=1..32.

Francisco J. Lopez, Francisco Martin, Complete minimal surfaces in R^3, April 11, 2000, web pdf, page 11.

Formula

p(x,t)=Exp(x*g*(t))*(1-f(t)^2);f(t)=1/Sqrt[1 - 14*t^4 + t^8];g(t)=t; p(x,t)=Sum[P(x,n)*t^n/n!,{n,0,Infinity}]; Out_n,m=(n!)*Coefficients(P(x,n).

Example

{0},
{0},
{0},
{0},
{-336},
{0, -1680},
{0, 0, -5040},
{0, 0, 0, -11760},
{-7862400, 0, 0, 0, -23520},
{0, -70761600, 0, 0, 0, -42336},
{0, 0, -353808000, 0, 0, 0, -70560}

Mathematica

Clear[p, f, g] g[t_] = t; f[t] = 1/Sqrt[1 - 14*t^4 + t^8]; p[t_] = Exp[x*g[t]]*(1 - f[t]^2); g = Table[ ExpandAll[(n!/3)*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]], {n, 0, 10}]; a = Table[ CoefficientList[(n!/3)*SeriesCoefficient[ FullSimplify[Series[p[t], {t, 0, 30}]], n], x], {n, 0, 10}]; Flatten[a]

Crossrefs

Sequence in context: A204612 A204796 A348822 * A090487 A289220 A060664

Adjacent sequences: A137519 A137520 A137521 * A137523 A137524 A137525

Keyword

uned,tabf,sign

Author

Roger L. Bagula, Apr 24 2008

Status

approved