login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Please make a donation (tax deductible in USA) to keep the OEIS running. Over 5000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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. 3
1, 6, 2268, 7434504, 95227613712, 3354162536029536, 264444869673131894208, 40740588107524550752746624, 11136881432872615930509713801472, 5026062205760019668688216299061782016 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Table of n, a(n) for n=0..9.

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 .

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 ;...

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). [From Paul D. Hanna, Jan 03 2009]

Sequence in context: A056048 A051113 A067174 * A059203 A254005 A198403

Adjacent sequences:  A153297 A153298 A153299 * A153301 A153302 A153303

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jan 02 2009

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified December 11 05:05 EST 2016. Contains 279034 sequences.