%I #25 Oct 23 2015 21:08:47
%S 1,-3,52,-1596,68174,-3679964,238949640,-18133397519,1578639190316,
%T -155623090726884,17203681850199360,-2116171636238243028,
%U 287762930191296817296,-43014624174283817327952,7032470676704382424751408,-1251802142595596587066746328,241602713767787669715442097616,-50368862903110844612768593045136,11303387910446267256159298807620472
%N G.f. A(x) satisfies: A(x) = B(x)^2 + C(x)^2 such that B(x) + I*C(x) = Series_Reversion(x - I*A(x)), where I^2 = -1.
%H Paul D. Hanna, <a href="/A263530/b263530.txt">Table of n, a(n) for n = 1..149</a>
%F Let G(x) be the g.f. of A141202, where G(x + G(x)*G(-x)) = x, and B(x) + I*C(x) = Series_Reversion(x - I*A(x)), then
%F (1) G(x)*G(-x) = A(I*x).
%F (2) G(x + A(I*x)) = x.
%F (3) G(x) = x - A( I*G(x) ).
%F (4) G(x) = -I*B(I*x) - C(I*x), where A(x) = B(x)^2 + C(x)^2.
%F (5) B(x) + I*C(x) = x - Sum_{n>=1} d^(n-1)/dx^(n-1) I^n*A(x)^n/n!, where A(x) = B(x)^2 + C(x)^2.
%e G.f.: A(x) = x^2 - 3*x^4 + 52*x^6 - 1596*x^8 + 68174*x^10 - 3679964*x^12 + 238949640*x^14 - 18133397519*x^16 +...
%e such that A(x) = B(x)^2 + C(x)^2 and B(x) and C(x) are defined by
%e Series_Reversion(x - I*A(x)) = B(x) + I*C(x), where
%e B(x) = x - 2*x^3 + 32*x^5 - 944*x^7 + 39366*x^9 - 2090576*x^11 + 134136792*x^13 - 10085875720*x^15 + 871536657504*x^17 +...+ (-1)^(n-1)*A141202(2*n-1)*x^(2*n-1) +...
%e C(x) = x^2 - 8*x^4 + 178*x^6 - 6255*x^8 + 293652*x^10 - 17085798*x^12 + 1182991528*x^14 - 95087538324*x^16 +...+ (-1)^(n-1)*A141202(2*n)*x^(2*n) +...
%e and
%e B(x)^2 = x^2 - 4*x^4 + 68*x^6 - 2016*x^8 + 83532*x^10 - 4399032*x^12 + 280046448*x^14 - 20916418480*x^16 + 1797498262020*x^18 +...
%e C(x)^2 = x^4 - 16*x^6 + 420*x^8 - 15358*x^10 + 719068*x^12 - 41096808*x^14 + 2783020961*x^16 - 218859071704*x^18 +...
%e Further
%e G(x) = -I*B(I*x) - C(I*x) = x + x^2 + 2*x^3 + 8*x^4 + 32*x^5 + 178*x^6 + 944*x^7 + 6255*x^8 + 39366*x^9 + 293652*x^10 +...+ A141202(n)*x^n +...
%e where G(x + G(x)*G(-x)) = x.
%o (PARI) {a(n) = my(A=x^2, D); for(i=0,2*n, D=serreverse(x - I*A +O(x^(2*n+1))); A = real(D)^2 + imag(D)^2 ); polcoeff(A,2*n)}
%o for(n=1,20,print1(a(n),", "))
%o (PARI) /* Differential Series */
%o {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
%o {a(n)=local(A = x^2 +O(x^(2*n+2))); for(i=1, 2*n, D = x + sum(m=1, 2*n, I^m*Dx(m-1, A^m/m!) +O(x^(2*n+2))); A = real(D)^2 + imag(D)^2 ); polcoeff(A, 2*n)}
%o for(n=1, 20, print1(a(n), ", "))
%Y Cf. A141202, A263531.
%K sign
%O 1,2
%A _Paul D. Hanna_, Oct 20 2015