G.f.: B(x) = (1/x)*Series_Reversion( x*(1 - 2*x)^2 / (1 - x^2) ).
G.f. B = B(x) and related functions A = A(x), C = C(x), D = D(x), satisfy:
(1.a) A = 1/((1 - 2*x*B)*(1 - 3*x*C)).
(1.b) B = 1/((1 - x*A)*(1 - 3*x*C)).
(1.c) C = 1/((1 - x*A)*(1 - 2*x*B)).
(1.d) D = 1/((1 - x*A)*(1 - 2*x*B)*(1 - 3*x*C)).
(1.e) D = sqrt(A*B*C).
(2.a) A = (1 + 2*x*D)*(1 + 3*x*D).
(2.b) B = (1 + x*D)*(1 + 3*x*D).
(2.c) C = (1 + x*D)*(1 + 2*x*D).
(2.d) D = (sqrt(24*A + 1) - 5)/(12*x) = (sqrt(12*B + 4) - 4)/(6*x) = (sqrt(8*C + 1) - 3)/(4*x).
(3.a) A = B/(1 - x*B) = C/(1 - 2*x*C) = D/(1 + x*D).
(3.b) B = C/(1 - x*C) = A/(1 + x*A) = D/(1 + 2*x*D).
(3.c) C = A/(1 + 2*x*A) = B/(1 + x*B) = D/(1 + 3*x*D).
(3.d) D = A/(1 - x*A) = B/(1 - 2*x*B) = C/(1 - 3*x*C).
(3.e) 1 = (1 + x*A)*(1 - x*B) = (1 + 2*x*A)*(1 - 2*x*C) = (1 + x*B)*(1 - x*C).
(3.f) 1 = (1 - x*A)*(1 + x*D) = (1 - 2*x*B)*(1 + 2*x*D) = (1 - 3*x*C)*(1 + 3*x*D).
(4.a) A = (1 + x*A)*(1 + 2*x*A)/(1 - x*A)^2.
(4.b) B = (1 - x^2*B^2)/(1 - 2*x*B)^2.
(4.c) C = (1 - x*C)*(1 - 2*x*C)/(1 - 3*x*C)^2.
(4.d) D = (1 + x*D)*(1 + 2*x*D)*(1 + 3*x*D).
(5.a) A = (1/x)*Series_Reversion( x*(1 - x)^2 / ((1 + x)*(1 + 2*x)) ).
(5.b) B = (1/x)*Series_Reversion( x*(1 - 2*x)^2 / (1 - x^2) ).
(5.c) C = (1/x)*Series_Reversion( x*(1 - 3*x)^2 / ((1 - x)*(1 - 2*x)) ).
(5.d) D = (1/x)*Series_Reversion( x / ((1 + x)*(1 + 2*x)*(1 + 3*x)) ).
a(n) ~ sqrt((2*s - 5*r*s^2 + 2*r^2*s^3) / (Pi*(22 - 8*r*s))) / (n^(3/2)*r^(n + 1/2)), where r = 0.07627811703169412709742160523783922642030319519275992338... and s = 2.2734876474240885308453076415547165781643109176277594587178... are positive real roots of the system of equations (1 - r^2*s^2)/(1 - 2*r*s)^2 = s, -1 + 8*r^3*s^3 - 2*r^2*s*(1 + 6*s) + r*(4 + 6*s) = 0. - Vaclav Kotesovec, Mar 02 2021
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