A120599 G.f. satisfies: 13*A(x) = 12 + 32*x + A(x)^5, starting with [1,4,20].

1, 4, 20, 280, 4660, 86728, 1727880, 36047280, 777470580, 17195957480, 387906427480, 8890184148560, 206419640698440, 4845319424269520, 114791477960006800, 2741248077305459040, 65915164046356799220

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..16.

Formula

G.f.: A(x) = 1 + Series_Reversion((1+13*x - (1+x)^5)/32). Lagrange Inversion yields: G.f.: A(x) = Sum_{n>=0} C(5*n,n)/(4*n+1) * (12+32*x)^(4*n+1)/13^(5*n+1). - Paul D. Hanna, Jan 24 2008

a(n) ~ 2^(-3/2 + 5*n) * (-12 + 4*(13/5)^(5/4))^(1/2 - n) / (5^(1/8) * 13^(3/8) * n^(3/2) * sqrt(Pi)). - Vaclav Kotesovec, Nov 28 2017

Example

A(x) = 1 + 4*x + 20*x^2 + 280*x^3 + 4660*x^4 + 86728*x^5 +...
A(x)^5 = 1 + 20*x + 260*x^2 + 3640*x^3 + 60580*x^4 + 1127464*x^5 +...

Mathematica

CoefficientList[1 + InverseSeries[Series[(1+13*x - (1+x)^5)/32, {x, 0, 20}], x], x] (* Vaclav Kotesovec, Nov 28 2017 *)

Prog

(PARI) {a(n)=local(A=1+4*x+20*x^2+x*O(x^n)); for(i=0, n, A=A+(-13*A+12+32*x+A^5)/8); polcoeff(A, n)}

Crossrefs

Cf. A120588 - A120598, A120600 - A120607.

Sequence in context: A120443 A053400 A362259 * A012797 A342907 A358544

Adjacent sequences: A120596 A120597 A120598 * A120600 A120601 A120602

Keyword

nonn

Author

Paul D. Hanna, Jun 16 2006

Status

approved