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G.f. satisfies: A(x) = Sum_{n>=0} x^n*(A(x)^n + A(-x)^n)/2.
2

%I #19 Jan 08 2023 19:59:03

%S 1,1,1,2,4,11,24,70,163,496,1199,3746,9291,29572,74712,241078,617612,

%T 2013979,5215990,17151362,44809646,148345585,390352720,1299558642,

%U 3440133086,11507046875,30615708092,102821083832,274759034372,925976586616,2483758221924

%N G.f. satisfies: A(x) = Sum_{n>=0} x^n*(A(x)^n + A(-x)^n)/2.

%C What is the limit a(n)^(1/n)?

%C Limit is equal to 3.16191277... - _Vaclav Kotesovec_, Jan 05 2023

%C Let 1/r equal the above limit, then A(r) = 1.9136996... and A(-r) = 0.7188624... - _Paul D. Hanna_, Jan 08 2023

%H Paul D. Hanna, <a href="/A196195/b196195.txt">Table of n, a(n) for n = 0..100</a>

%F G.f. A(x) satisfies:

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

%F (2) 0 = (x - 3) + (-4*x + 7)*A(x) + (5*x^2 - 4)*A(x)^2 + (-12*x^2 + 4*x)*A(x)^3 + (4*x^3 + 4*x^2)*A(x)^4 + (x^4 - 4*x^3)*A(x)^5.

%F (3) 0 = A^5*x^4 + 4*(1 - A)*A^4*x^3 + (5 - 12*A + 4*A^2)*A^2*x^2 + (1 - 4*A + 4*A^3)*x - (3 - 7*A + 4*A^2), where A = A(x). - _Paul D. Hanna_, Jan 08 2023

%F (4) x*A(-x) = (2 - (2+x)*A(x) + 2*x*A(x)^2) / (1 - 2*A(x) + 2*x*A(x)^2). - _Paul D. Hanna_, Jan 08 2023

%e G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 4*x^4 + 11*x^5 + 24*x^6 + 70*x^7 + ...

%e where

%e A(x) = 1 + x*(A(x) + A(-x))/2 + x^2*(A(x)^2 + A(-x)^2)/2 + x^3*(A(x)^3 + A(-x)^3)/2 + x^4*(A(x)^4 + A(-x)^4)/2 + x^5*(A(x)^5 + A(-x)^5)/2 + ...

%e Related expansions begin:

%e (A(x) + A(-x))/2 = 1 + x^2 + 4*x^4 + 24*x^6 + 163*x^8 + 1199*x^10 + ...

%e (A(x)^2 + A(-x)^2)/2 = 1 + 3*x^2 + 13*x^4 + 82*x^6 + 574*x^8 + ...

%e (A(x)^3 + A(-x)^3)/2 = 1 + 6*x^2 + 30*x^4 + 199*x^6 + 1443*x^8 + ...

%e (A(x)^4 + A(-x)^4)/2 = 1 + 10*x^2 + 59*x^4 + 414*x^6 + 3121*x^8 + ...

%e (A(x)^5 + A(-x)^5)/2 = 1 + 15*x^2 + 105*x^4 + 785*x^6 + 6170*x^8 + ...

%e (A(x)^6 + A(-x)^6)/2 = 1 + 21*x^2 + 174*x^4 + 1395*x^6 + 11460*x^8 + ...

%e ...

%e G.f. A = A(x) is a solution to the 5th degree polynomial equation:

%e 0 = (x - 3) + (7 - 4*x)*A - (4 - 5*x^2)*A^2 + 4*(1 - 3*x)*x*A^3 + 4*(1 + x)*x^2*A^4 - (4 - x)*x^3*A^5.

%o (PARI) {a(n)=local(A=1+x); for(k=1, n, A=(1/(1-x*A+x*O(x^n))+1/(1-x*subst(A,x,-x+x*O(x^n))))/2); polcoeff(A, n)}

%o for(n=0,30, print1(a(n),", "))

%o (PARI) {a(n)=local(A=1+x+x*O(x^n)); for(k=1, n, A=1+sum(j=1, n, x^j*(A^j+subst(A, x,-x)^j)/2)); polcoeff(A, n)}

%o for(n=0,30, print1(a(n),", "))

%Y Cf. A196196.

%K nonn

%O 0,4

%A _Paul D. Hanna_, Sep 29 2011