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A141202
G.f. satisfies: A(x + A(x)*A(-x)) = x.
4
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
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
Sequence in context: A051636 A081561 A009753 * A081358 A294506 A206303
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