OFFSET
1,4
COMMENTS
Equals the series reversion of the g.f. of A265940.
FORMULA
Let B(x) be the series reversion of A(x) so that A(B(x)) = x, then
(1) A(x^2) = x*B(x) - x*B(x)^2.
(2) B( x*B(x) - x*B(x)^2 ) = x^2.
(3) B(x) = (1 - sqrt(1 - 4*A(x^2)/x)) / 2.
(4) B(x) = C( A(x^2)/x ), where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers A000108.
EXAMPLE
G.f.: A(x) = x - x^2 + x^3 - 3*x^4 + 5*x^5 - 11*x^6 + 27*x^7 - 69*x^8 + 187*x^9 - 517*x^10 + 1461*x^11 - 4163*x^12 + 11947*x^13 + ...
such that A(A(x)^2) = (x-x^2)*A(x) where
A(A(x)^2) = x^2 - 2*x^3 + 2*x^4 - 4*x^5 + 8*x^6 - 16*x^7 + 38*x^8 - 96*x^9 + 256*x^10 - 704*x^11 + 1978*x^12 - 5624*x^13 + 16110*x^14 + ...
Let B(x) be the series reversion of A(x) so that A(B(x)) = x then
B(x) = x + x^2 + x^3 + 3*x^4 + 9*x^5 + 25*x^6 + 71*x^7 + 219*x^8 + 689*x^9 + 2189*x^10 + 7059*x^11 + 23091*x^12 + ... + A265940(n)*x^n + ...
such that B( x*B(x) - x*B(x)^2 ) = x^2.
PROG
(PARI) {a(n) = my(A=x); for(i=1, #binary(n), A = serreverse( (1 - sqrt(1 - 4*subst(A +x*O(x^n), x, x^2)/x) )/2 ) ); polcoeff(A, n)}
for(n=1, 40, print1(a(n), ", "))
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Dec 29 2015
STATUS
approved