0,2

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 .

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 ;...

(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))}

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

Sequence in context: A013723 A159405 A060617 this_sequence A129042 A123401 A079303

Adjacent sequences: A153298 A153299 A153300 this_sequence A153302 A153303 A153304

nonn

Paul D. Hanna (pauldhanna(AT)juno.com), Jan 02 2009