A177408 - OEIS

%I #2 Mar 30 2012 18:37:21

%S 1,2,8,40,224,1352,8576,56352,380160,2617584,18320384,129950912,

%T 932114432,6749344832,49268899840,362189529344,2678989406208,

%U 19923485019840,148887398711296,1117452514604800,8419605676818432

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

%F Radius of convergence, r, and related values:

%F . r = 0.123195593008501117935531659506400229201428882504980293279833...

%F . A(r) = 0.239702251488238187695726754757078686233527461098463854580...

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

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

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

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

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

%e G.f.: A(x) = x + 2*x^2 + 8*x^3 + 40*x^4 + 224*x^5 + 1352*x^6 +...

%e Related expansions:

%e . A(4A(x)^4) = 4*x^4 + 32*x^5 + 224*x^6 + 1536*x^7 + 10592*x^8 +...

%e . A(x)^4 = x^4 + 8*x^5 + 56*x^6 + 384*x^7 + 2640*x^8 + 18336*x^9 +...

%e . A(4x^4)^(1/2) = 2*x^2 + 8*x^6 + 112*x^10 + 2112*x^14 + 45760*x^18 +...

%e ...

%e The series reversion is defined by R(x) = x - A(4x^4)^(1/2) where:

%e . R(x) = x - 2*x^2 - 8*x^6 - 112*x^10 - 2112*x^14 - 45760*x^18 -...

%e . x/R(x) = 1 + 2*x + 4*x^2 + 8*x^3 + 16*x^4 + 40*x^5 + 96*x^6 + 224*x^7 +...

%o (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)}

%Y Cf. A141200.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Jun 19 2010