A153301 Coefficient of x^(4n+1)/(4n+1)! in the Maclaurin expansion of sm4(x), which is a generalization of the Dixon elliptic function sm(x,0) defined by A104133.

1, 18, 14364, 70203672, 1192064637456, 52269828456672288, 4930307288899134335424, 884135650165992118901204352, 275721138550891190637445080842496

Offset

0,2

Comments

Equals column 0 of triangle A357800.

Links

Table of n, a(n) for n=0..8.

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 cm4(x)^4 = 1 + sm4(x)^4, where cm4(x) is the g.f. of A153300, then:

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

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

Example

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

Mathematica

With[{n = 8}, CoefficientList[Series[JacobiSN[Sqrt[2] x^(1/4), 1/2]/(x^(1/4) Sqrt[2 JacobiCN[Sqrt[2] x^(1/4), 1/2]]), {x, 0, n}], x] Table[(4 k + 1)!, {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=intformal((1+intformal(A^3))^3)); n=4*n+1; n!*polcoeff(A, n))}

Crossrefs

Cf. A104133; A153300, A153302 (cm4(x)^2 + sm4(x)^2).

Cf. A357800.

Sequence in context: A060617 A222202 A201986 * A129042 A262359 A202155

Adjacent sequences: A153298 A153299 A153300 * A153302 A153303 A153304

Keyword

nonn

Author

Paul D. Hanna, Jan 02 2009

Status

approved