OFFSET
0,3
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..370
FORMULA
E.g.f.: 1/(1 - 2*Series_Reversion( Integral (1 - 4*x^2)^(3/2) dx ))^(1/2).
Limit n->infinity (a(n)/n!)^(1/n) = 32/(3*Pi) = 3.3953054526271... - Vaclav Kotesovec, Jan 29 2014
a(n) ~ n! * 2^(3/10) * (32/(3*Pi))^(n+1/5) / (GAMMA(1/5) * 5^(1/5) * n^(4/5)). - Vaclav Kotesovec, Jan 30 2014
EXAMPLE
E.g.f.: A(x) = 1 + x + 3*x^2/2! + 27*x^3/3! + 249*x^4/4! + 4041*x^5/5! +...
Related series.
A(x)^6 = 1 + 6*x + 48*x^2/2! + 552*x^3/3! + 8064*x^4/4! + 146016*x^5/5! +...
Note that 1 - 1/A(x)^2 is an odd function:
1 - 1/A(x)^2 = 2*x + 24*x^3/3! + 2592*x^5/5! + 768384*x^7/7! +...
where Series_Reversion((1 - 1/A(x)^2)/2) = Integral (1-4*x^2)^(3/2) dx.
MATHEMATICA
CoefficientList[1/(1 - 2*InverseSeries[Series[Integrate[(1 - 4*x^2)^(3/2), x], {x, 0, 20}], x])^(1/2), x] * Range[0, 20]! (* Vaclav Kotesovec, Jan 28 2014 *)
PROG
(PARI) {a(n)=local(A=1); for(i=0, n, A=1+intformal(A^6*subst(A, x, -x)^3 +x*O(x^n) )); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n)=local(A=1); A=1/(1-2*serreverse(intformal((1-4*x^2 +x*O(x^n))^(3/2))))^(1/2); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 07 2014
STATUS
approved