OFFSET
0,3
COMMENTS
This sequence consists entirely of positive integers, but negative coefficients appear in G(x) where G(x) = 1 + x*G(x)^2 + x^2/G(x)^4.
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..400
Vaclav Kotesovec, Recurrence (of order 8)
FORMULA
G.f. satisfies: x^2 = A(x)^4 - A(x)^3 - x*A(x)^5.
a(n) ~ s*sqrt((2*r + s^5) / (Pi*(6*r + s^5))) / (2*n^(3/2)*r^n), where r = 0.2481615785281025222129020972009425192154180157568... and s = 2.036459583599244077761952153978454195241402482624... are roots of the system of equations 1 + r^2/s^3 + r*s^2 = s, 2*r*s = 1 + 3*r^2/s^4. - Vaclav Kotesovec, Nov 18 2017
EXAMPLE
G.f.: A(x) = 1 + x + 3*x^2 + 4*x^3 + 11*x^4 + 53*x^5 + 146*x^6 +...
Related expansions begin:
A(x)^2 = 1 + 2*x + 7*x^2 + 14*x^3 + 39*x^4 + 152*x^5 + 480*x^6 +...
1/A(x)^3 = 1 - 3*x - 3*x^2 + 14*x^3 - 6*x^4 - 114*x^5 + 133*x^6 +...
PROG
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+x*A^2+x^2/(A+x*O(x^n))^3); polcoeff(A, n)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 07 2011
STATUS
approved