A177408 G.f. satisfies: A(x) = x + A( 4*A(x)^4 )^(1/2).

1, 2, 8, 40, 224, 1352, 8576, 56352, 380160, 2617584, 18320384, 129950912, 932114432, 6749344832, 49268899840, 362189529344, 2678989406208, 19923485019840, 148887398711296, 1117452514604800, 8419605676818432

Offset

1,2

Links

Table of n, a(n) for n=1..21.

Formula

Radius of convergence, r, and related values:

. r = 0.123195593008501117935531659506400229201428882504980293279833...

. A(r) = 0.239702251488238187695726754757078686233527461098463854580...

. A(-r) = -0.1022686661772839286606841162458831990656192887231153817...

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

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

. R(x) = x - A(4x^4)^(1/2),

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

Example

G.f.: A(x) = x + 2*x^2 + 8*x^3 + 40*x^4 + 224*x^5 + 1352*x^6 +...
Related expansions:
. A(4A(x)^4) = 4*x^4 + 32*x^5 + 224*x^6 + 1536*x^7 + 10592*x^8 +...
. A(x)^4 = x^4 + 8*x^5 + 56*x^6 + 384*x^7 + 2640*x^8 + 18336*x^9 +...
. A(4x^4)^(1/2) = 2*x^2 + 8*x^6 + 112*x^10 + 2112*x^14 + 45760*x^18 +...
...
The series reversion is defined by R(x) = x - A(4x^4)^(1/2) where:
. R(x) = x - 2*x^2 - 8*x^6 - 112*x^10 - 2112*x^14 - 45760*x^18 -...
. x/R(x) = 1 + 2*x + 4*x^2 + 8*x^3 + 16*x^4 + 40*x^5 + 96*x^6 + 224*x^7 +...

Prog

(PARI) {a(n)=local(A=x+x^2); for(i=1, n, A=x+subst(A, x, 4*(A+x*O(x^n))^4)^(1/2)); polcoeff(A, n)}

Crossrefs

Cf. A141200.

Sequence in context: A214760 A052701 A151374 * A289431 A337912 A085485

Adjacent sequences: A177405 A177406 A177407 * A177409 A177410 A177411

Keyword

nonn

Author

Paul D. Hanna, Jun 19 2010

Status

approved