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A263530 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. 3
1, -3, 52, -1596, 68174, -3679964, 238949640, -18133397519, 1578639190316, -155623090726884, 17203681850199360, -2116171636238243028, 287762930191296817296, -43014624174283817327952, 7032470676704382424751408, -1251802142595596587066746328, 241602713767787669715442097616, -50368862903110844612768593045136, 11303387910446267256159298807620472 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Paul D. Hanna, Table of n, a(n) for n = 1..149

FORMULA

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

(1) G(x)*G(-x) = A(I*x).

(2) G(x + A(I*x)) = x.

(3) G(x) = x - A( I*G(x) ).

(4) G(x) = -I*B(I*x) - C(I*x), where A(x) = B(x)^2 + C(x)^2.

(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.

EXAMPLE

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 +...

such that A(x) = B(x)^2 + C(x)^2 and B(x) and C(x) are defined by

Series_Reversion(x - I*A(x)) = B(x) + I*C(x), where

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) +...

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) +...

and

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 +...

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 +...

Further

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 +...

where G(x + G(x)*G(-x)) = x.

PROG

(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)}

for(n=1, 20, print1(a(n), ", "))

(PARI) /* Differential Series */

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}

{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)}

for(n=1, 20, print1(a(n), ", "))

CROSSREFS

Cf. A141202, A263531.

Sequence in context: A030180 A005547 A301948 * A136723 A202649 A302932

Adjacent sequences:  A263527 A263528 A263529 * A263531 A263532 A263533

KEYWORD

sign

AUTHOR

Paul D. Hanna, Oct 20 2015

STATUS

approved

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Last modified December 8 12:31 EST 2019. Contains 329864 sequences. (Running on oeis4.)