A120598 G.f. satisfies: 30*A(x) = 29 + 125*x + A(x)^5, starting with [1,5,10].

1, 5, 10, 90, 825, 8445, 92820, 1066740, 12670635, 154308775, 1916370170, 24177471370, 309007779015, 3992428316835, 52059968802000, 684240882022800, 9055282215370050, 120563388411386850, 1613785688724362400

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

Table of n, a(n) for n=0..18.

Formula

G.f.: A(x) = 1 + Series_Reversion((1+30*x - (1+x)^5)/125). Lagrange Inversion yields: G.f.: A(x) = Sum_{n>=0} C(5*n,n)/(4*n+1) * (29+125*x)^(4*n+1)/30^(5*n+1). - Paul D. Hanna, Jan 24 2008

a(n) ~ 5^(-1/2 + 3*n) * (-29 + 24*6^(1/4))^(1/2 - n) / (2^(15/8) * 3^(3/8) * n^(3/2) * sqrt(Pi)). - Vaclav Kotesovec, Nov 28 2017

Example

A(x) = 1 + 5*x + 10*x^2 + 90*x^3 + 825*x^4 + 8445*x^5 +...
A(x)^5 = 1 + 25*x + 300*x^2 + 2700*x^3 + 24750*x^4 + 253350*x^5 +...

Mathematica

CoefficientList[1 + InverseSeries[Series[(1+30*x - (1+x)^5)/125, {x, 0, 20}], x], x] (* Vaclav Kotesovec, Nov 28 2017 *)

Prog

(PARI) {a(n)=local(A=1+5*x+10*x^2+x*O(x^n)); for(i=0, n, A=A+(-30*A+29+125*x+A^5)/25); polcoeff(A, n)}

Crossrefs

Cf. A120588 - A120597, A120599 - A120607.

Sequence in context: A111992 A325583 A344194 * A200983 A344140 A267042

Adjacent sequences: A120595 A120596 A120597 * A120599 A120600 A120601

Keyword

nonn

Author

Paul D. Hanna, Jun 16 2006

Status

approved