A395427 Coefficient of x^(3n)/(3n)! in the Maclaurin expansion of a function g(x,0) associated with the Dixon elliptic functions.

1, 1, -2, -8, -152, 1240, -6848, -621440, -8103296, -515196800, 22483912960, -34790666240, -8062284861440, 1844661863895040, 196434444070666240, 1027011084429721600, 532650564250569441280, -94237121671171272704000, 2039228675045199496806400, 513714003355389609017344000, -5209573728611533514689740800

Offset

0,3

Comments

Following Dixon, let g(x) = g(x,0) be the function whose logarithmic derivative is f(x) = f(x,0), where f(x,0) is the integral of sm(x,0) * cm(x,0), and sm and cm are the Dixon elliptic functions. g(x) can be expressed in terms of Weierstrass elliptic functions as g(x)=((1/3 - p'(x ; 0, 1/27)) * sigma(x ; 0, 1/27)^3)/2.

References

A. C. Dixon, On the doubly periodic functions arising out of the curve x^3 + y^3 - 3 alpha xy=1, Quart. J. Pure Appl. Math. 24 (1890), 167-233.

Links

Table of n, a(n) for n=0..20.

Formula

E.g.f.: Sum_{k>=0} a(k) * x^(3*k) / (3*k)! = g(x).

Example

g(x) = 1 + 1/6 * x^3 - 1/360 * x^6 - 1/45360 * x^9 - 19/59875200 * x^12 + ...

Mathematica

Table[With[{m = 3 n}, m! SeriesCoefficient[(1/3 - WeierstrassPPrime[x, {0, 1/27}]) WeierstrassSigma[x, {0, 1/27}]^3/2, {x, 0, m}]], {n, 0, 20}]

Crossrefs

Cf. A104133, A104134 (Dixon elliptic functions).

Cf. A381359 (associated Dixon function f).

Sequence in context: A091299 A362729 A307326 * A007314 A102099 A012298

Adjacent sequences: A395424 A395425 A395426 * A395428 A395429 A395430

Keyword

sign

Author

Jan Mangaldan, Apr 23 2026

Status

approved