A368626 - OEIS

%I #17 Jan 10 2024 23:57:59

%S 1,1,2,9,22,138,356,2585,6830,53838,144156,1197546,3233692,27859444,

%T 75665736,669553209,1826204958,16493851110,45131989100,414263198030,

%U 1136416283860,10568504182860,29050963193720,273107307342090,751985844723308,7133921326564172,19670502565821464

%N Expansion of g.f. A(x) satisfying A(x) = 1 + x*(A(x)^2 - A(-x)^2)/2 + x*(A(x)^3 + A(-x)^3)/2.

%C Conjecture: a(n) is odd when n = 2^k - 1 for k >= 0 and even elsewhere.

%H Paul D. Hanna, <a href="/A368626/b368626.txt">Table of n, a(n) for n = 0..600</a>

%F G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.

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

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

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

%F (4.a) A(x) = (1 - sqrt(1-8*x + 4*x*A(-x) + 4*x^2*A(-x)^2)) / (2*x).

%F (4.b) A(-x) = (sqrt(1+8*x - 4*x*A(x) + 4*x^2*A(x)^2) - 1) / (2*x).

%F (5) (A(x) + A(-x))/2 = 1/(1 - 2*x*(A(x) - A(-x))/2).

%e G.f.: A(x) = 1 + x + 2*x^2 + 9*x^3 + 22*x^4 + 138*x^5 + 356*x^6 + 2585*x^7 + 6830*x^8 + 53838*x^9 + 144156*x^10 + 1197546*x^11 + 3233692*x^12 + ...

%e where A(x) is formed from the odd bisection of A(x)^2 and the even bisection of A(x)^3, as can be seen from the expansions

%e A(x)^2 = 1 + 2*x + 5*x^2 + 22*x^3 + 66*x^4 + 356*x^5 + 1157*x^6 + 6830*x^7 + 23222*x^8 + 144156*x^9 + 504546*x^10 + ...

%e A(x)^3 = 1 + 3*x + 9*x^2 + 40*x^3 + 138*x^4 + 693*x^5 + 2585*x^6 + 13764*x^7 + 53838*x^8 + 296646*x^9 + 1197546*x^10 + ...

%e so that the bisections of the above series are related by

%e (A(x) + A(-x))/2 = 1 + x*(A(x)^2 - A(-x)^2)/2, and

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

%e SPECIFIC VALUES.

%e A(t) = 3/2 at t = 0.1819737010113140094420890735437063355509087658723835...

%e with A(-t) = 0.7945570310255352575261389299040205708629421553742768...

%e G.f. A(x) diverges at x = 1/5.4, but converges at x = 1/5.5 to yield

%e A(1/5.5) = 1.496543384376249917206500686071412596234401473798923...

%e A(-1/5.5) = 0.795582249398671834477410218197255634423553817319574...

%e Other values are as follows.

%e A(1/6) = 1.34228124014121938629204994980825043322418782558714594...

%e A(-1/6) = 0.84031658679173656850293071643280362490543801455743768...

%e A(1/7) = 1.23812032178413019856840253750104622400159644919325618...

%e A(-1/7) = 0.87219621912499007272745977375746581998964690903627574...

%e A(1/8) = 1.18723993315598647777707954645984780429075497185978705...

%e A(-1/8) = 0.88995083754758616465388572384122362483578619460668827...

%o (PARI) {a(n) = my(A=1+x, A_); for(i=1, n, A=truncate(A) + x*O(x^i); B=subst(A,x,-x); A = 1 + x*(A^2 - B^2)/2 + x*(A^3 + B^3)/2 ; ); polcoeff(A,n)}

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

%Y Cf. A368627.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jan 09 2024