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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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified March 20 22:57 EDT 2023. Contains 361392 sequences. (Running on oeis4.)