A228867 G.f. A(x) satisfies: A(x) = -x + x^2 + A(A(x)) + Series_Reversion(A(A(x))).

1, 1, 8, 140, 3536, 111118, 4067460, 167387778, 7579673514, 372383647086, 19656714142024, 1107002518847134, 66161264225994340, 4179178067931209524, 278086164995822234072, 19439660328872258046471, 1424349675827697250143308, 109166829826333936529736762

Offset

1,3

Comments

Compare to the Catalan function C(x) that (trivially) satisfies:

C(x) = -x + x^2 + C(x) + Series_Reversion(C(x)).

Links

Table of n, a(n) for n=1..18.

Example

G.f.: A(x) = x + x^2 + 8*x^3 + 140*x^4 + 3536*x^5 + 111118*x^6 +...
where
A(A(x)) = x + 2*x^2 + 18*x^3 + 321*x^4 + 8144*x^5 + 256404*x^6 + 9395688*x^7 + 386935778*x^8 + 17530417564*x^9 +...
Let the series reversion of A(x) be denoted by G(x), which begins
G(x) = x - x^2 - 6*x^3 - 105*x^4 - 2658*x^5 - 83608*x^6 - 3062080*x^7 -...
then
G(G(x)) = x - 2*x^2 - 10*x^3 - 181*x^4 - 4608*x^5 - 145286*x^6 - 5328228*x^7 - 219548000*x^8 - 9950744050*x^9 +...
so that A(x) = -x + x^2 + A(A(x)) + G(G(x)).

Prog

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

Crossrefs

Sequence in context: A234353 A212442 A185248 * A224735 A090931 A367199

Adjacent sequences: A228864 A228865 A228866 * A228868 A228869 A228870

Keyword

nonn

Author

Paul D. Hanna, Sep 06 2013

Status

approved