OFFSET
1,2
COMMENTS
Compare to F( F(x)^2 ) = x*F(x)/(1 - F(x)) when F(x) = x/(1-x).
Conjecture: a(2^k + 1) == 2 (mod 4) for k >= 0, and a(n) == 0 (mod 4) for n > 0 when n is not of the form 2^k + 1.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..520
FORMULA
G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) A( A(x)^2 ) = x*A(x)/(1 - 2*A(x)).
(2) A( A(x^2)*(1 - 2*x)/x ) = x.
(3) A( x^2*A(x)^2/(1 - 2*A(x))^2 ) = x*A(x)^3/(1 - 2*(1+x)*A(x)).
EXAMPLE
G.f.: A(x) = x + 2*x^2 + 6*x^3 + 24*x^4 + 110*x^5 + 544*x^6 + 2824*x^7 + 15168*x^8 + 83566*x^9 + 469568*x^10 + 2680496*x^11 + 15500000*x^12 + ...
where A( A(x)^2 ) = x*A(x)/(1 - 2*A(x)).
RELATED SERIES.
Series_Reversion( A(x) ) = x - 2*x^2 + 2*x^3 - 4*x^4 + 6*x^5 - 12*x^6 + 24*x^7 - 48*x^8 + 110*x^9 - 220*x^10 + 544*x^11 - 1088*x^12 + ...
which equals A(x^2)/x - 2*A(x^2).
The square of the g.f. A(x) starts
A(x)^2 = x^2 + 4*x^3 + 16*x^4 + 72*x^5 + 352*x^6 + 1816*x^7 + 9720*x^8 + 53440*x^9 + 299904*x^10 + 1710648*x^11 + 9887336*x^12 + ...
where
A( A(x)^2 ) = x^2 + 4*x^3 + 18*x^4 + 88*x^5 + 454*x^6 + 2432*x^7 + 13392*x^8 + 75312*x^9 + 430606*x^10 + 2495232*x^11 + 14619672*x^12 + ...
which equals x*A(x)/(1 - 2*A(x)).
SPECIFIC VALUES.
Radius of convergence r satisfies A(r) = 1/3 at r = A(1/9) given below.
A(1/7) = 0.241813310611551220817899738183178164198414429806944245427...
A(1/8) = 0.183769291658204237933331788389613939490616376081980660279...
A(1/9) = 0.152116846037417910191393667087080967002339480852805927801...
A(1/16) = 0.072296818077433380544156938042004242365657632791493646470...
A(1/25) = 0.043659522607225680741528652737350659693714843830834115952...
A(t) = 1/4 at t = 0.144593636154866761088313876084008484731315265582987...
A(t) = 1/5 at t = 0.130978567821677042224585958212051979081144531492502...
PROG
(PARI) {a(n) = my(A=[0, 1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff( x*Ser(A)/(1 - 2*Ser(A)) - subst(Ser(A), x, Ser(A)^2 ), #A) ); A[n+1]}
for(n=1, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, May 13 2024
STATUS
approved