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+15*x - (1+x)^6)/27). Lagrange Inversion yields: G.f.: A(x) = Sum_{n>=0} C(6*n,n)/(5*n+1) * (14+27*x)^(5*n+1)/15^(6*n+1). - Paul D. Hanna, Jan 24 2008
a(n) ~ 3^(-1/2 + 3*n) * (-14 + 5*(5/2)^(6/5))^(1/2 - n) / (2^(3/5) * 5^(9/10) * n^(3/2) * sqrt(Pi)). - Vaclav Kotesovec, Nov 28 2017
EXAMPLE
A(x) = 1 + 3*x + 15*x^2 + 210*x^3 + 3510*x^4 + 65562*x^5 +...
A(x)^6 = 1 + 18*x + 225*x^2 + 3150*x^3 + 52650*x^4 + 983430*x^5 +...
MATHEMATICA
CoefficientList[1 + InverseSeries[Series[(1+15*x - (1+x)^6)/27, {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+(-15*A+14+27*x+A^6)/9); polcoeff(A, n)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 16 2006
STATUS
approved