OFFSET
0,3
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..390
Vaclav Kotesovec, Recurrence (of order 8)
FORMULA
E.g.f.: 1/(1 - Series_Reversion( Integral (1-x^2)^4 dx )).
a(n) ~ n! * 2^(4/5) * (315/128)^(n+1/5) / (Gamma(1/5) * 5^(1/5) * n^(4/5)). - Vaclav Kotesovec, Jan 29 2014
EXAMPLE
E.g.f.: A(x) = 1 + x + 2*x^2/2! + 14*x^3/3! + 88*x^4/4! + 1096*x^5/5! +...
Related series.
A(x)^6 = 1 + 6*x + 42*x^2/2! + 384*x^3/3! + 4368*x^4/4! + 60096*x^5/5! +...
Note that 1 - 1/A(x) is an odd function:
1 - 1/A(x) = x + 8*x^3/3! + 496*x^5/5! + 81728*x^7/7! +...
where Series_Reversion(1 - 1/A(x)) = Integral (1-x^2)^4 dx.
MATHEMATICA
CoefficientList[1/(1 - InverseSeries[Series[Integrate[(1-x^2)^4, x], {x, 0, 20}], x]), 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)^4 +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^2 +x*O(x^n))^4))); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 07 2014
STATUS
approved