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%I
%S 1,1,1,2,4,9,21,50,122,302,758,1928,4958,12849,33509,88122,233810,
%T 621022,1641150,4411180,12364368,33073210,71807506,206985492,
%U 1354944972,3153779248,-33794258600,-62697691948,2524565441138,5004344042337,-186642439700891,-368380986364150,16196862324254354,32039943659306982,-1602823227559245434
%N G.f. A(x) satisfies: A( A( x - x^2 ) ) = x + x^2.
%C a(2*n-1) = A277292(n).
%C 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).
%H Paul D. Hanna, <a href="/A318008/b318008.txt">Table of n, a(n) for n = 1..300</a>
%F G.f. A(x) satisfies:
%F (1) A(-A(-x)) = x.
%F (2a) A(A(x)) = 2*C(x) - x, where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).
%F (2b) A(A( x - x^2 )) = x + x^2.
%F (2c) A(A( x/(1+x)^2 )) = (x + 2*x^2)/(1+x)^2.
%F (3a) (A(x) - A(-x))^2 = 2*(A(x) + A(-x)).
%F (3b) A(x)^2 - 2*A(x) - 2*A(x)*A(-x) + A(-x)^2 - 2*A(-x) = 0.
%F Define B(x) = (A(x) - A(-x))/2 and Catalan series C(x) = x + C(x)^2, then
%F (4a) B(x)^2 = (A(x) + A(-x))/2.
%F (4b) A(x) = B(x) + B(x)^2.
%F (5a) B( A(x - x^2) ) = x.
%F (5b) B( A(x) ) = C(x).
%F (6a) A( B(x) - B(x)^2 ) = x.
%F (6b) B( B(x) + B(x)^2 ) = C(x).
%F (6c) C( B(x) - B(x)^2 ) = B(x).
%e 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 + ...
%e such that A(A(x - x^2)) = x + x^2.
%e RELATED SERIES.
%e (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 + ...
%e (b) The odd bisection B(x) = (A(x) - A(-x))/2 begins
%e 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) + ...
%e such that B(x)^2 yields the even bisection (A(x) + A(-x))/2
%e 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) + ...
%e thus A(x) = B(x) + B(x)^2.
%e (c) Also, the Catalan series equals
%e 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 + ...
%e (d) Note that A(x - x^2) equals the series reversion of B(x):
%e 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 + ...
%e where B( A(x -x^2) ) = x and A( B(x) - B(x)^2 ) = x.
%o (PARI) /* Using A(A( x - x^2 )) = x + x^2. */
%o {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)}
%o for(n=1,30,print1(a(n),", "))
%Y Cf. A277292, A000108.
%K sign
%O 1,4
%A _Paul D. Hanna_, Sep 06 2018
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