OFFSET
0,2
COMMENTS
See comments in A120588 for conditions needed for an integer sequence to satisfy a functional equation of the form: r*A(x) = c + b*x + A(x)^n.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..70
FORMULA
G.f.: A(x) = 1 + Series_Reversion((1+5*x - (1+x)^3)/4).
G.f.: A(x) = Sum_{n>=0} C(3*n,n)/(2*n+1) * (4+4*x)^(2*n+1) / 5^(3*n+1), due to Lagrange Inversion.
Recurrence: 17*(n-1)*n*a(n) = 108*(n-1)*(2*n-3)*a(n-1) + 12*(3*n-7)*(3*n-5)*a(n-2). - Vaclav Kotesovec, Oct 19 2012
a(n) ~ sqrt(250-60*sqrt(15))*((108+30*sqrt(15))/17)^n/(30*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 19 2012
EXAMPLE
A(x) = 1 + 2*x + 6*x^2 + 40*x^3 + 330*x^4 + 3048*x^5 + 30156*x^6 +...
A(x)^3 = 1 + 6*x + 30*x^2 + 200*x^3 + 1650*x^4 +15240*x^5 +150780*x^6 +...
MATHEMATICA
FullSimplify[Table[SeriesCoefficient[Sum[Binomial[3*k, k]/(2*k+1)*(4+4*x)^(2*k+1)/5^(3*k+1), {k, 0, Infinity}], {x, 0, n}], {n, 0, 20}]] (* Vaclav Kotesovec, Oct 19 2012 *)
PROG
(PARI) {a(n)=local(A=1+2*x+6*x^2+x*O(x^n)); for(i=0, n, A=A+(-5*A+4+4*x+A^3)/2); polcoeff(A, n)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 16 2006, Jan 24 2008
STATUS
approved