A318008 G.f. A(x) satisfies: A( A( x - x^2 ) ) = x + x^2.

1, 1, 1, 2, 4, 9, 21, 50, 122, 302, 758, 1928, 4958, 12849, 33509, 88122, 233810, 621022, 1641150, 4411180, 12364368, 33073210, 71807506, 206985492, 1354944972, 3153779248, -33794258600, -62697691948, 2524565441138, 5004344042337, -186642439700891, -368380986364150, 16196862324254354, 32039943659306982, -1602823227559245434

Offset: 1

Author

Paul D. Hanna, Sep 06 2018

Keywords

• sign

Comments

a(2*n-1) = A277292(n).

a(2^k-1) = 1 (mod 2) and a(2^(k+1)-2) = 1 (mod 2) for k >= 1, and a(n) is even elsewhere (conjecture).

Examples

			G.f.: A(x) = x + x^2 + x^3 + 2*x^4 + 4*x^5 + 9*x^6 + 21*x^7 + 50*x^8 + 122*x^9 + 302*x^10 + 758*x^11 + 1928*x^12 + 4958*x^13 + 12849*x^14 + 33509*x^15 + ...
such that A(A(x - x^2)) = x + x^2.
RELATED SERIES.
(a) A(A(x)) = x + 2*x^2 + 4*x^3 + 10*x^4 + 28*x^5 + 84*x^6 + 264*x^7 + 858*x^8 + 2860*x^9 + 9724*x^10 + ... + 2*A000108(n-2)*x^n + ...
(b) The odd bisection B(x) = (A(x) - A(-x))/2 begins
B(x) = x + x^3 + 4*x^5 + 21*x^7 + 122*x^9 + 758*x^11 + 4958*x^13 + 33509*x^15 + 233810*x^17 + 1641150*x^19 + 12364368*x^21 + ... + A277292(n)*x^(2*n-1) + ...
such that B(x)^2 yields the even bisection (A(x) + A(-x))/2
B(x)^2 = x^2 + 2*x^4 + 9*x^6 + 50*x^8 + 302*x^10 + 1928*x^12 + 12849*x^14 + 88122*x^16 + 621022*x^18 + 4411180*x^20 + ... + a(2*n)*x^(2*n) + ...
thus A(x) = B(x) + B(x)^2.
(c) Also, the Catalan series equals
B( B(x) + B(x)^2 ) = x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 + 42*x^6 + 132*x^7 + 429*x^8 + 1430*x^9 + 4862*x^10 + ... + A000108(n-2)*x^n + ...
(d) Note that A(x - x^2) equals the series reversion of B(x):
A(x - x^2) = x - x^3 - x^5 - x^7 + 3*x^9 + 11*x^11 - 9*x^13 + 71*x^15 - 1685*x^17 + 31683*x^19 - 845729*x^21 + 28968319*x^23 + ...
where B( A(x  -x^2) ) = x and A( B(x) - B(x)^2 ) = x.
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..300

Crossrefs

Cf. A277292, A000108.

Programs

PARI

/* Using A(A( x - x^2 )) = x + x^2. */
{a(n) = my(A=x+x*O(x^n)); for(i=1,n, A = A + (x+x^2 - subst(A,x,subst(A,x,x-x^2)) )/2 ); polcoeff(A,n)}
for(n=1,30,print1(a(n),", "))

Formula

G.f. A(x) satisfies:

(1) A(-A(-x)) = x.

(2a) A(A(x)) = 2*C(x) - x, where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).

(2b) A(A( x - x^2 )) = x + x^2.

(2c) A(A( x/(1+x)^2 )) = (x + 2*x^2)/(1+x)^2.

(3a) (A(x) - A(-x))^2 = 2*(A(x) + A(-x)).

(3b) A(x)^2 - 2*A(x) - 2*A(x)*A(-x) + A(-x)^2 - 2*A(-x) = 0.

Define B(x) = (A(x) - A(-x))/2 and Catalan series C(x) = x + C(x)^2, then

(4a) B(x)^2 = (A(x) + A(-x))/2.

(4b) A(x) = B(x) + B(x)^2.

(5a) B( A(x - x^2) ) = x.

(5b) B( A(x) ) = C(x).

(6a) A( B(x) - B(x)^2 ) = x.

(6b) B( B(x) + B(x)^2 ) = C(x).

(6c) C( B(x) - B(x)^2 ) = B(x).