A263531 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.

1, -5, 112, -4320, 227766, -14942616, 1162657840, -104338906529, 10609887976616, -1207797487940348, 152572977202977992, -21242819435887437760, 3241842130718219392320, -539712032454499745200600, 97612800729251959183577168, -19106581507633892101354812324, 4033513580481891302243479168168, -915408408852469072798058443048672

Offset

1,2

Links

Paul D. Hanna, Table of n, a(n) for n = 1..150

Formula

Let G(x) be the g.f. of A227852, where G( x - (G(x)^2 + G(-x)^2)/2 ) = x, and B(x) + I*C(x) = Series_Reversion(x - I*A(x)), then

(1) (G(x)^2 + G(-x)^2)/2 = -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 - 5*x^4 + 112*x^6 - 4320*x^8 + 227766*x^10 - 14942616*x^12 + 1162657840*x^14 - 104338906529*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 + 44*x^5 - 1728*x^7 + 93130*x^9 - 6235288*x^11 + 493813936*x^13 - 44989814920*x^15 + 4633862094852*x^17 +...+ (-1)^(n-1)*A227852(2*n-1)*x^(2*n-1) +...
C(x) = x^2 - 10*x^4 + 294*x^6 - 13389*x^8 + 796620*x^10 - 57551130*x^12 + 4857378920*x^14 - 468103507718*x^16 +...+ (-1)^(n-1)*A227852(2*n)*x^(2*n)
and
B(x)^2 = x^2 - 4*x^4 + 92*x^6 - 3632*x^8 + 195108*x^10 - 12995160*x^12 + 1023750448*x^14 - 92825448208*x^16 + 9521361427980*x^18 +...
C(x)^2 = x^4 - 20*x^6 + 688*x^8 - 32658*x^10 + 1947456*x^12 - 138907392*x^14 + 11513458321*x^16 - 1088526548636*x^18 +...
Further
G(x) = -I*B(I*x) - C(I*x) = x + x^2 + 2*x^3 + 10*x^4 + 44*x^5 + 294*x^6 + 1728*x^7 + 13389*x^8 + 93130*x^9 + 796620*x^10 +...+ A227852(n)*x^n +...
where G( x - (G(x)^2 + G(-x)^2)/2 ) = 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. A227852, A263530.

Sequence in context: A002400 A268404 A258795 * A351148 A258177 A224897

Adjacent sequences: A263528 A263529 A263530 * A263532 A263533 A263534

Keyword

sign

Author

Paul D. Hanna, Oct 20 2015

Status

approved