A177406 G.f. satisfies: A(x) = x + A( 27*A(x)^6 )^(1/3).

1, 3, 18, 135, 1134, 10206, 96228, 938304, 9384660, 95746860, 992583072, 10425704562, 110714749236, 1186711306875, 12821975547696, 139501306797120, 1527013735182810, 16805125811826495, 185831030179447380

Offset

1,2

Links

Table of n, a(n) for n=1..19.

Formula

Radius of convergence, r, and related values:

. r = 0.0832854117848379079627858177662093190328717029025025344504328...

. A(r) = 0.166285097718710273401082966562979331796241671228716865630919...

. limit a(n)/a(n+1) = r.

Series reversion: let R(x) satisfy R(A(x)) = x, then

. R(x) = x - A(27x^6)^(1/3),

. x/R(x) = x*d/dx[x/R(x)] at x = A(r) where r = radius of convergence.

Example

G.f. A(x) = x + 3*x^2 + 18*x^3 + 135*x^4 + 1134*x^5 + 10206*x^6 +...
Related expansions:
. A(27*A(x)^6) = 27*x^6 + 486*x^7 + 6561*x^8 + 80190*x^9 +...
. A(x)^6 = x^6 + 18*x^7 + 243*x^8 + 2970*x^9 + 34749*x^10 +...
. A(27*x^6)^(1/3) = 3*x^2 + 18*x^3 + 135*x^4 + 1134*x^5 + 10206*x^6 +...
...
The series reversion is defined by R(x) = x - A(27x^6)^(1/3) where:
. R(x) = x - 3*x^2 - 81*x^8 - 10935*x^14 - 2047032*x^20 -...
. x/R(x) = 1 + 3*x + 9*x^2 + 27*x^3 + 81*x^4 + 243*x^5 + 729*x^6 + 2268*x^7 +...

Prog

(PARI) {a(n)=local(A=x+x^2); for(i=1, n, A=x+subst(A, x, 27*(A+x*O(x^n))^6)^(1/3)); polcoeff(A, n)}

Crossrefs

Cf. A177408.

Sequence in context: A114178 A005159 A151383 * A289430 A247452 A371416

Adjacent sequences: A177403 A177404 A177405 * A177407 A177408 A177409

Keyword

nonn

Author

Paul D. Hanna, Jun 20 2010

Status

approved