A372527 - OEIS

%I #13 May 14 2024 00:42:00

%S 1,2,6,24,110,544,2824,15168,83566,469568,2680496,15500000,90596880,

%T 534368880,3176515904,19010408448,114444697214,692566900736,

%U 4210541805680,25704699739296,157508966594744,968415775598608,5972400758691808,36936064677976320,229015813606251672

%N Expansion of g.f. A(x) satisfying A( A(x)^2 ) = x*A(x)/(1 - 2*A(x)).

%C Compare to F( F(x)^2 ) = x*F(x)/(1 - F(x)) when F(x) = x/(1-x).

%C 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.

%H Paul D. Hanna, <a href="/A372527/b372527.txt">Table of n, a(n) for n = 1..520</a>

%F G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.

%F (1) A( A(x)^2 ) = x*A(x)/(1 - 2*A(x)).

%F (2) A( A(x^2)*(1 - 2*x)/x ) = x.

%F (3) A( x^2*A(x)^2/(1 - 2*A(x))^2 ) = x*A(x)^3/(1 - 2*(1+x)*A(x)).

%e 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 + ...

%e where A( A(x)^2 ) = x*A(x)/(1 - 2*A(x)).

%e RELATED SERIES.

%e 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 + ...

%e which equals A(x^2)/x - 2*A(x^2).

%e The square of the g.f. A(x) starts

%e 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 + ...

%e where

%e 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 + ...

%e which equals x*A(x)/(1 - 2*A(x)).

%e SPECIFIC VALUES.

%e Radius of convergence r satisfies A(r) = 1/3 at r = A(1/9) given below.

%e A(1/7) = 0.241813310611551220817899738183178164198414429806944245427...

%e A(1/8) = 0.183769291658204237933331788389613939490616376081980660279...

%e A(1/9) = 0.152116846037417910191393667087080967002339480852805927801...

%e A(1/16) = 0.072296818077433380544156938042004242365657632791493646470...

%e A(1/25) = 0.043659522607225680741528652737350659693714843830834115952...

%e A(t) = 1/4 at t = 0.144593636154866761088313876084008484731315265582987...

%e A(t) = 1/5 at t = 0.130978567821677042224585958212051979081144531492502...

%o (PARI) {a(n) = my(A=[0,1]); for(i=1,n, A = concat(A,0);

%o A[#A] = polcoeff( x*Ser(A)/(1 - 2*Ser(A)) - subst(Ser(A),x, Ser(A)^2 ), #A) );A[n+1]}

%o for(n=1,30,print1(a(n),", "))

%K nonn

%O 1,2

%A _Paul D. Hanna_, May 13 2024