OFFSET
0,3
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..400
FORMULA
G.f. satisfies: A(x) = 1 - x*(A(x) - A(x)^2 - A(x)^3) + x^2*(A(x) - 2*A(x)^2 - A(x)^4) + x^3*A(x)^4.
a(n) ~ sqrt((1/(-1 + r)^2 - s/(-1 + r*s)^2 + s^2/(-1 + r*s^2)^2) / (Pi*(1/(-1 + r*s^2)^2 + r*(1/(-1 + r*s)^3 - (4*s^2) / (-1 + r*s^2)^3)))) / (2*n^(3/2)*r^n), where r = 0.1741099691155951761402154753241071226265020289369... and s = 1.469614426933947254586622522985062658500679266649... are roots of the system of equations 1/(1-r) + 1/(-1+r*s) + 1/(1-r*s^2) = s, 2*r*s / (-1+r*s^2)^2 = 1 + r/(-1+r*s)^2. - Vaclav Kotesovec, Nov 18 2017
EXAMPLE
G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 23*x^4 + 98*x^5 + 440*x^6 +...
Related series begin:
1/(1-x*A(x)) = 1 + x + 2*x^2 + 5*x^3 + 15*x^4 + 53*x^5 + 211*x^6 +...
1/(1-x*A(x)^2) = 1 + x + 3*x^2 + 10*x^3 + 37*x^4 + 150*x^5 + 650*x^6 +...
PROG
(PARI) {a(n)=local(A=1+x, X=x+x*O(x^n)); for(i=1, n, A=1/(1-X)-1/(1-X*A)+1/(1-X*A^2)); polcoeff(A, n)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 26 2011
STATUS
approved