OFFSET
0,3
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..380
FORMULA
E.g.f.: 1/sqrt(1 - 2*Series_Reversion( Integral 1 - 4*x^2 dx )).
E.g.f.: 1/(1 - Series_Reversion( Integral (1-x)^3*(1+2*x-x^2) dx )).
a(n) ~ n! * GAMMA(3/4) * 3^(n+1/4) / (2^(3/4) * Pi * n^(3/4)) * (1 + sqrt(3)*Pi / (72*sqrt(n)*GAMMA(3/4)^2) - (-1)^n*3^(1/4)*2^(3/4)*sqrt(Pi) / (24*n^(3/4)*GAMMA(3/4))). - Vaclav Kotesovec, Jan 27 2014
EXAMPLE
E.g.f.: A(x) = 1 + x + 3*x^2/2! + 23*x^3/3! + 201*x^4/4! + 2785*x^5/5! +...
Related series.
A(x)^2 = 1 + 2*x + 8*x^2/2! + 64*x^3/3! + 640*x^4/4! + 8960*x^5/5! +...
A(x)^5 = 1 + 5*x + 35*x^2/2! + 355*x^3/3! + 4585*x^4/4! + 73445*x^5/5! +...
Note that 1 - 1/A(x)^2 is an odd function that begins:
1 - 1/A(x)^2 = 2*x + 16*x^3/3! + 1280*x^5/5! + 286720*x^7/7! + 126156800*x^9/9! +...
such that Series_Reversion( (1 - 1/A(x)^2)/2 ) = x - 4*x^3/3.
MATHEMATICA
CoefficientList[1/Sqrt[1-2*InverseSeries[Series[x-4*x^3/3, {x, 0, 20}], x]], x]*Range[0, 20]! (* Vaclav Kotesovec, Jan 27 2014 *)
PROG
(PARI) /* By definition A'(x) = A(x)^5 * A(-x)^2: */
{a(n)=local(A=1); for(i=0, n, A=1+intformal(A^5*subst(A, x, -x)^2 +x*O(x^n) )); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n)=local(A=1); A=1/sqrt(1-2*serreverse(intformal(1-4*x^2 +x*O(x^n) ))); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n)=local(A=1); A=1/(1 -serreverse( intformal((1-x)^3*(1+2*x-x^2) +x*O(x^n) ))); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 07 2014
STATUS
approved