A177749 G.f. satisfies: A(A(A(x))) = (1 + A(x))^2 - (1+x).

1, 1, -4, 44, -702, 13890, -319068, 8214003, -231978050, 7087571850, -232015437564, 8081329261812, -297945065180148, 11580691120595856, -473034996747412360, 20252620565496493579, -906876693691788342040

Offset

1,3

Links

Table of n, a(n) for n=1..17.

Formula

G.f. satisfies: A(A(x)) = 2*x + x^2 - A^{-1}(x), where A^{-1}(x) denotes the series reversion of A(x).

Example

G.f.: A(x) = x + x^2 - 4*x^3 + 44*x^4 - 702*x^5 + 13890*x^6 -+...
Compare the series expansions:
A(A(A(x))) = x + 3*x^2 - 6*x^3 + 81*x^4 - 1324*x^5 + 26480*x^6 +...
(1+A(x))^2 = 1 + 2*x +3*x^2 -6*x^3 +81*x^4 -1324*x^5 +26480*x^6 +...
to see that A(A(A(x))) = (1 + A(x))^2 - (1+x).
Also compare:
A(A(x)) = x +2*x^2 -6*x^3 +69*x^4 -1112*x^5 +22094*x^6 -508720*x^7 +...
A^{-1}(x) = x -x^2 +6*x^3 -69*x^4 +1112*x^5 -22094*x^6 +508720*x^7 -...
to see that A(A(x)) + A^{-1}(x) = 2*x + x^2.

Prog

(PARI) {a(n)=local(F=x+x^2+sum(m=3, n-1, a(m)*x^m)+x*O(x^n)); if(n<1, 0, if(n<3, 1, -polcoeff(subst(F, x, F)+serreverse(F), n)))}

Crossrefs

Sequence in context: A371680 A056063 A218224 * A291198 A053332 A276369

Adjacent sequences: A177746 A177747 A177748 * A177750 A177751 A177752

Keyword

sign

Author

Paul D. Hanna, May 13 2010

Status

approved