OFFSET
0,3
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..358
FORMULA
E.g.f.: 1/(1 - 4*Series_Reversion( Integral (1 - 16*x^2)^(1/4) dx ))^(1/4).
Lim_{n->infinity} (a(n)/n!)^(1/n) = 12*sqrt(2) * GAMMA(3/4)^2 / Pi^(3/2) = 4.57655858101... - Vaclav Kotesovec, Jan 28 2014
EXAMPLE
E.g.f.: A(x) = 1 + x + 5*x^2/2! + 53*x^3/3! + 745*x^4/4! + 14761*x^5/5! + ...
Related series.
A(x)^6 = 1 + 6*x + 60*x^2/2! + 888*x^3/3! + 17040*x^4/4! + 407136*x^5/5! + ...
Note that 1 - 1/A(x)^4 is an odd function:
1 - 1/A(x)^4 = 4*x + 32*x^3/3! + 4864*x^5/5! + 2250752*x^7/7! + ...
where Series_Reversion((1 - 1/A(x)^4)/4) = Integral (1-16*x^2)^(1/4) dx.
MATHEMATICA
CoefficientList[1/(1 - 4*InverseSeries[Series[Integrate[(1-16*x^2)^(1/4), x], {x, 0, 20}], x])^(1/4), 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)^1 +x*O(x^n) )); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n)=local(A=1); A=1/(1-4*serreverse(intformal((1-16*x^2 +x*O(x^n))^(1/4))))^(1/4); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 07 2014
STATUS
approved