A141202 G.f. satisfies: A(x + A(x)*A(-x)) = x.

1, 1, 2, 8, 32, 178, 944, 6255, 39366, 293652, 2090576, 17085798, 134136792, 1182991528, 10085875720, 95087538324, 871536657504, 8727880568468, 85385942061016, 904071273001352, 9389429908430784, 104728235042891360, 1149676904405092704, 13467595558130095308, 155705728677310569008

Offset

1,3

Links

Paul D. Hanna, Table of n, a(n) for n = 1..300

Formula

G.f. A(x) satisfies:

(1) A(x) = x - A(-A(x)) * A(A(x)).

(2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) (-A(x)*A(-x))^n / n!.

(3) A(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) (-A(x)*A(-x))^n / (n!*x) ).

(4) A(-I*x) * A(I*x) = F(x), where F(x) is the g.f. of A263530 and satisfies: F(x) = B(x)^2 + C(x)^2 such that B(x) + I*C(x) = Series_Reversion(x - I*F(x)), where I^2 = -1.

Example

G.f.: A(x) = x + x^2 + 2*x^3 + 8*x^4 + 32*x^5 + 178*x^6 + 944*x^7 +...
By definition, Series_Reversion(A(x)) = x + A(-x)*A(x) where
A(-x)*A(x) = -x^2 - 3*x^4 - 52*x^6 - 1596*x^8 - 68174*x^10 - 3679964*x^12 +...+ (-1)^n * A263530(n)*x^(2*n) +...
Consequently, A(x) = x - A(-A(x))*A(A(x)) where
-A(-A(x)) = x + 0*x^2 + 2*x^3 + x^4 + 30*x^5 + 38*x^6 + 852*x^7 +...
A(A(x)) = x + 2*x^2 + 6*x^3 + 27*x^4 + 134*x^5 + 786*x^6 + 4852*x^7 +...
The related g.f. of A263530, F(x) = A(-I*x)*A(I*x), satisfies: F(x) = B(x)^2 + C(x)^2 such that B(x) + I*C(x) = Series_Reversion(x - I*F(x)), where I^2 = -1:
F(x) = x^2 - 3*x^4 + 52*x^6 - 1596*x^8 + 68174*x^10 - 3679964*x^12 +...

Mathematica

m = 26; A[_] = 0;
Do[A[x_] = x - A[-A[x]] A[A[x]] + O[x]^m // Normal, {m}];
CoefficientList[A[x]/x, x] (* Jean-François Alcover, Oct 01 2019 *)

Prog

(PARI) {a(n)=local(A=x+x^2); for(i=0, n, A=serreverse(x+A*subst(A, x, -x+x*O(x^n)))) ; polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))
(PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); for(i=1, n, A=x+sum(m=1, n, Dx(m-1, (A*subst(-A, x, -x))^m/m!))+x*O(x^n)); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A263530, A227852.

Sequence in context: A051636 A081561 A009753 * A081358 A294506 A206303

Adjacent sequences: A141199 A141200 A141201 * A141203 A141204 A141205

Keyword

nonn

Author

Paul D. Hanna, Jun 13 2008, Sep 05 2008

Extensions

Edited by N. J. A. Sloane, Sep 13 2008 at the suggestion of R. J. Mathar

Status

approved