OFFSET
0,3
FORMULA
E.g.f.: 1/(1 - 7*Series_Reversion( Integral 1/(1 - 49*x^2)^(1/7) dx ))^(1/7).
EXAMPLE
E.g.f.: A(x) = 1 + x + 8*x^2/2! + 106*x^3/3! + 2192*x^4/4! + 57016*x^5/5! +...
Related series.
A(x)^7 = 1 + 7*x + 98*x^2/2! + 1960*x^3/3! + 52136*x^4/4! + 1709512*x^5/5! +...
Note that 1 - 1/A(x)^7 is an odd function:
1 - 1/A(x)^7 = 7*x - 98*x^3/3! - 19208*x^5/5! - 14137088*x^7/7! -...
where Series_Reversion((1 - 1/A(x)^7)/7) = Integral 1/(1-49*x^2)^(1/7) dx.
PROG
(PARI) /* By definition A'(x) = A(x)^7 / A(-x): */
{a(n)=local(A=1); for(i=0, n, A=1+intformal(A^7/subst(A, x, -x) +x*O(x^n) )); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n)=local(A=1); A=1/(1-7*serreverse(intformal(1/(1-49*x^2 +x*O(x^n))^(1/7))))^(1/7); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Apr 09 2014
STATUS
approved