OFFSET
1,3
COMMENTS
Compare to the following related identities.
Let F(x) = x*C(x^2) such that C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108), then F(x) satisfies:
(1) Series_Reversion( F(x)/(1 - i*F(x) + F(x)^2) ) = F(x)/(1 + i*F(x) + F(x)^2),
(2) Series_Reversion( F(x)/(1 - i*x + x*F(x)) ) = F(x)/(1 + i*x + x*F(x)),
in which the inverse of the composite function equals the conjugate of that function.
FORMULA
G.f. A(x) satisfies: A(x)^2 = x*B1(x)/B2(x) - 1, where B1(x) + i*B2(x) = A(x)/(1 - i*x + A(x)^2).
EXAMPLE
G.f.: A(x) = x + x^3 + 3*x^5 + 12*x^7 + 59*x^9 + 365*x^11 + 3088*x^13 + 38996*x^15 + 740467*x^17 + 19758435*x^19 +...
such that the inverse of function A(x)/(1-i*x + A(x)^2) equals its conjugate.
RELATED SERIES.
A(x)^2 = x^2 + 2*x^4 + 7*x^6 + 30*x^8 + 151*x^10 + 920*x^12 + 7404*x^14 + 87774*x^16 + 1589695*x^18 + 41348962*x^20 +...
Let A(x)/(1 - i*x + A(x)^2) = B1(x) + i*B2(x),
A(x)/(1 - i*x + A(x)^2) = x + i*x^2 - x^3 - 2*i*x^4 + 3*x^5 + 3*i*x^6 - 4*x^7 - 8*i*x^8 + 11*x^9 + 9*i*x^10 - 13*x^11 - 35*i*x^12 + 46*x^13 + 13*i*x^14 - 26*x^15 - 193*i*x^16 + 239*x^17 - 177*i*x^18 + 151*x^19 - 1472*i*x^20 + 1711*x^21 - 3328*i*x^22 + 3479*x^23 - 14789*i*x^24 + 16500*x^25 - 45614*i*x^26 + 49093*x^27 - 173708*i*x^28 + 190208*x^29 - 598306*i*x^30 +...
then the series reversion of B1(x) + i*B2(x) equals the conjugate B1(x) - i*B2(x).
Note that A(x)^2 = x*B1(x)/B2(x) - 1, which holds because
B1(x) = A(x)*(1 + A(x)^2)/((1 + A(x)^2)^2 + x^2), and
B2(x) = A(x)*x/((1 + A(x)^2)^2 + x^2).
PROG
(PARI) {a(n) = my(Oxn=x*O(x^(2*n)), A = x +Oxn); for(i=1, 2*n, A = A + (x - subst(A/(1+I*x + A^2), x, A/(1-I*x + A^2) ))/2); polcoeff(A, 2*n-1)}
for(n=1, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 10 2016
STATUS
approved