A153300 Coefficient of x^(4n)/(4n)! in the Maclaurin expansion of cm4(x), which is a generalization of the Dixon elliptic function cm(x,0) defined by A104134.

1, 6, 2268, 7434504, 95227613712, 3354162536029536, 264444869673131894208, 40740588107524550752746624, 11136881432872615930509713801472, 5026062205760019668688216299061782016

Offset

0,2

Comments

Equals column 0 of triangle A357801.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..117

Alessandro Gambini, Giorgio Nicoletti, and Daniele Ritelli, The Wallis Products for Fermat Curves, Vietnam J. Math. (2023).

Alessandro Gambini, Giorgio Nicoletti, and Daniele Ritelli, A Structural Approach to Gudermannian Functions, Results in Mathematics (2024) Vol. 79, Art. No. 10.

Formula

Define sm4(x)^4 = cm4(x)^4 - 1, where sm4(x) is the g.f. of A153301, then:

d/dx cm4(x) = sm4(x)^3 ;

d/dx sm4(x) = cm4(x)^3 .

a(n) ~ 2^(14*n + 11/4) * Gamma(3/4)^(8*n+1) * n^(4*n) / (exp(4*n) * Pi^(6*n + 3/4)). - Vaclav Kotesovec, Oct 06 2019

Example

G.f.: cm4(x) = 1 + 6*x^4/4! + 2268*x^8/8! + 7434504*x^12/12! + 95227613712*x^16/16! +...
sm4(x) = x + 18*x^5/5! + 14364*x^9/9! + 70203672*x^13/13! + 1192064637456*x^17/17! +...
These functions satisfy: cm4(x)^4 - sm4(x)^4 = 1 where:
cm4(x)^4 = 1 + 24*x^4/4! + 24192*x^8/8! + 140507136*x^12/12! + 2716743794688*x^16/16 +...
RELATED EXPANSIONS:
cm4(x)^2 = 1 + 12*x^4/4! + 7056*x^8/8! + 28340928*x^12/12! + 419025809664*x^16/16! +...
sm4(x)^2 = 2*x^2/2! + 216*x^6/6! + 368928*x^10/10! + 3000945024*x^14/14! +...
cm4(x)^3 = 1 + 18*x^4/4! + 14364*x^8/8! + 70203672*x^12/12! + 1192064637456*x^16/16! +...
sm4(x)^3 = 6*x^3/3! + 2268*x^7/7! + 7434504*x^11/11! + 95227613712*x^15/15! +...
DERIVATIVES:
d/dx cm4(x) = sm4(x)^3 ;
d^2/dx^2 cm4(x) = 3*cm4(x)^3*sm4(x)^2 ;
d^3/dx^3 cm4(x) = 6*cm4(x)^6*sm4(x) + 9*cm4(x)^2*sm4(x)^5 ;
d^4/dx^4 cm4(x) = 6*cm4(x)^9 + 81*cm4(x)^5*sm4(x)^4 + 18*cm4(x)*sm4(x)^8 ;...

Mathematica

With[{n = 9}, CoefficientList[Series[JacobiDN[Sqrt[2] x^(1/4), 1/2]/Sqrt[JacobiCN[Sqrt[2] x^(1/4), 1/2]], {x, 0, n}], x] Table[(4 k)!, {k, 0, n}]] (* Jan Mangaldan, Jan 04 2017 *)

Prog

(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x); for(i=0, n, A=1+intformal(intformal(A^3)^3)); n=4*n; n!*polcoeff(A, n))}

Crossrefs

Cf. A104134; A153301, A153302 (cm4(x)^2 + sm4(x)^2).

Cf. A153303 (cm4(x)^4), A357801.

Sequence in context: A056048 A051113 A067174 * A059203 A254005 A279654

Adjacent sequences: A153297 A153298 A153299 * A153301 A153302 A153303

Keyword

nonn

Author

Paul D. Hanna, Jan 02 2009

Status

approved