OFFSET
1,4
COMMENTS
Let (R_n) be the sequence of rational functions satisfying: R_1(x) = 1; R_{n+1}(x) = (R_n(x) * 2*x/(1+x^2))'. By definition, a(n) = R_n(1).
Applying [Dominici, Theorem 4.1] proves that the e.g.f. of this sequence is the series reversion of log(1+x)/2 + x^2/4 + x/2.
LINKS
Luc Rousseau, Table of n, a(n) for n = 1..300
D. Dominici, Nested derivatives: A simple method for computing series expansions of inverse functions, arXiv:math/0501052 [math.CA], 2005.
FORMULA
EXAMPLE
R_1(x) = 1,
so a(1) = R_1(1) = 1.
R_2(x) = (R_1(x)*2*x/(1+x^2))' = (1 * 2*x/(1+x^2))' = 2*(1-x^2)/(1+x^2)^2,
so a(2) = R_2(1) = 0.
R_3(x) = (R_2(x)*2*x/(1+x^2))' = (2*(1-x^2)/(1+x^2)^2 * 2*x/(1+x^2))' = 4*(1-8*x^2+3*x^4)/(1+x^2)^4, so a(3) = R_3(1) = -1.
PROG
(PARI)
list_a(nmax)=my(n, r); n=1; r=1; print1(subst(r, x, 1), ", "); while(n<nmax, n++; r=(r*2*x/(1+x^2))'; print1(subst(r, x, 1), ", "))
list_a(50)
(PARI) my(x='x+O('x^33)); Vec(serlaplace(serreverse(log(1+x)/2 + x^2/4 + x/2))) \\ Joerg Arndt, Dec 22 2020
CROSSREFS
KEYWORD
sign
AUTHOR
Luc Rousseau, Dec 07 2020
STATUS
approved