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 A227852 G.f. A(x) satisfies: A(x) = Series_Reversion( x - (A(x)^2 + A(-x)^2)/2 ). 3
 1, 1, 2, 10, 44, 294, 1728, 13389, 93130, 796620, 6235288, 57551130, 493813936, 4857378920, 44989814920, 468103507718, 4633862094852, 50749496457992, 533271010341720, 6126256486912776, 67990630238066888, 817168635245112432, 9541543704324657008, 119719059789052412360 (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))^2 + A(-A(x))^2)/2. (2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) (A(x)^2 + A(-x)^2)^n / (n!*2^n). (3) (A(I*x)^2 + A(-I*x)^2)/2 = -F(x), where F(x) is the g.f. of A263531 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 + 10*x^4 + 44*x^5 + 294*x^6 + 1728*x^7 +... The series reversion of A(x), G(x) where A(G(x)) = x, begins: G(x) = x - x^2 - 5*x^4 - 112*x^6 - 4320*x^8 - 227766*x^10 - 14942616*x^12 - 1162657840*x^14 +...+ (-1)^n * A263531(n)*x^(2*n) +... and can be formed from a bisection of A(x)^2: A(x)^2 = x^2 + 2*x^3 + 5*x^4 + 24*x^5 + 112*x^6 + 716*x^7 + 4320*x^8 + 32290*x^9 + 227766*x^10 + 1893488*x^11 + 14942616*x^12 + 134816212*x^13 + 1162657840*x^14 +... The related g.f. of A263531, F(x) = -(A(I*x)^2 + A(-I*x)^2)/2, 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 - 5*x^4 + 112*x^6 - 4320*x^8 + 227766*x^10 - 14942616*x^12 +... PROG (PARI) {a(n)=local(A=x); for(i=1, n, A=serreverse(x-(A^2+subst(A^2, x, -x +x*O(x^n)))/2)); polcoeff(A, n)} for(n=1, 25, 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^2+subst(A, x, -x)^2)^m/2^m/m!))+x*O(x^n)); polcoeff(A, n)} for(n=1, 25, print1(a(n), ", ")) CROSSREFS Cf. A263531, A213591, A141202. Sequence in context: A151313 A144896 A142983 * A347524 A065805 A145239 Adjacent sequences:  A227849 A227850 A227851 * A227853 A227854 A227855 KEYWORD nonn AUTHOR Paul D. Hanna, Oct 31 2013 STATUS approved

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Last modified June 30 03:34 EDT 2022. Contains 354913 sequences. (Running on oeis4.)