A214404 G.f. satisfies: A( x - A(x)^2 ) = x+x^2 - A(x)^2.

1, 1, 2, 9, 48, 296, 2008, 14648, 113200, 917588, 7746876, 67770456, 611916624, 5685473544, 54227943240, 529937718704, 5297716934498, 54106608947506, 563945862248108, 5993092373220992, 64885877599868336, 715222369910418672, 8021722347464144744

Offset

1,3

Links

Table of n, a(n) for n=1..23.

Formula

G.f. satisfies: A(x) = x + G(x)^2 where G(x - A(x)^2) = x.

Example

G.f.: A(x) = x + x^2 + 2*x^3 + 9*x^4 + 48*x^5 + 296*x^6 + 2008*x^7 +...
Related expansions:
A(x)^2 = x^2 + 2*x^3 + 5*x^4 + 22*x^5 + 118*x^6 + 724*x^7 + 4881*x^8 +...
A(x-A(x)^2) = x - 2*x^3 - 5*x^4 - 22*x^5 - 118*x^6 - 724*x^7 -...
x+x^2 - A(x)^2 = x - 2*x^3 - 5*x^4 - 22*x^5 - 118*x^6 - 724*x^7 -...
Let G(x) equal the series reversion of x - A(x)^2:
G(x) = x + x^2 + 4*x^3 + 20*x^4 + 120*x^5 + 804*x^6 + 5840*x^7 +...
then
G(x)^2 = x^2 + 2*x^3 + 9*x^4 + 48*x^5 + 296*x^6 + 2008*x^7 + 14648*x^8 +...
A(G(x)) = x + 2*x^2 + 8*x^3 + 44*x^4 + 282*x^5 + 2004*x^6 + 15340*x^7 +...
A(G(x))^2 = x^2 + 4*x^3 + 20*x^4 + 120*x^5 + 804*x^6 + 5840*x^7 +...
where A(x) = x + G(x)^2 = G(x) + G(x)^2 - A(G(x))^2.

Prog

(PARI) {a(n)=local(A=x+x^2); for(i=1, n, A=x+serreverse(x-A^2+x*O(x^n))^2); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A216171.

Sequence in context: A171803 A368961 A100427 * A074143 A198892 A357790

Adjacent sequences: A214401 A214402 A214403 * A214405 A214406 A214407

Keyword

nonn

Author

Paul D. Hanna, Jul 15 2012

Status

approved