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A141202
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G.f. satisfies: A(x + A(x)*A(-x)) = x.
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4
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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
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OFFSET
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1,3
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LINKS
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Paul D. Hanna, Table of n, a(n) for n = 1..300
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FORMULA
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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.
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EXAMPLE
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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 +...
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MATHEMATICA
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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 *)
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PROG
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(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), ", "))
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CROSSREFS
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Cf. A263530, A227852.
Sequence in context: A051636 A081561 A009753 * A081358 A294506 A206303
Adjacent sequences: A141199 A141200 A141201 * A141203 A141204 A141205
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KEYWORD
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nonn
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AUTHOR
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Paul D. Hanna, Jun 13 2008, Sep 05 2008
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EXTENSIONS
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Edited by N. J. A. Sloane, Sep 13 2008 at the suggestion of R. J. Mathar
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STATUS
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approved
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