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.
FORMULA
G.f.: A(x) = 1 + Series_Reversion((1+37*x - (1+x)^10)/81). Lagrange Inversion yields: G.f.: A(x) = Sum_{n>=0} C(9*n,n)/(8*n+1) * (36+81*x)^(8*n+1)/37^(9*n+1). - Paul D. Hanna, Jan 24 2008
a(n) ~ 3^(-1 + 4*n) * (-36 + 9*(37/10)^(10/9))^(1/2 - n) / (2^(5/9) * 5^(1/18) * 37^(4/9) * n^(3/2) * sqrt(Pi)). - Vaclav Kotesovec, Nov 28 2017
EXAMPLE
A(x) = 1 + 3*x + 15*x^2 + 270*x^3 + 5505*x^4 + 124818*x^5 +...
A(x)^10 = 1 + 30*x + 555*x^2 + 9990*x^3 + 203685*x^4 + 4618266*x^5 +...
MATHEMATICA
CoefficientList[1 + InverseSeries[Series[(1+37*x - (1+x)^10)/81, {x, 0, 20}], x], x] (* Vaclav Kotesovec, Nov 28 2017 *)
PROG
(PARI) {a(n)=local(A=1+3*x+15*x^2+x*O(x^n)); for(i=0, n, A=A+(-37*A+36+81*x+A^10)/27); polcoeff(A, n)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 16 2006
STATUS
approved