A368627 - OEIS

%I #27 Jan 22 2024 12:47:13

%S 1,1,3,7,40,103,723,1941,15060,41382,340657,950061,8132676,22916139,

%T 201684153,572618987,5145063940,14692661910,134152006842,384852888898,

%U 3559210821120,10248531332559,95777105998365,276630878235275,2607824127882204,7551545042631558,71714198513326425

%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.

%H Paul D. Hanna, <a href="/A368627/b368627.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) = A(-x) + x*A(x)^2 + x*A(-x)^2.

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

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

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

%e G.f.: A(x) = 1 + x + 3*x^2 + 7*x^3 + 40*x^4 + 103*x^5 + 723*x^6 + 1941*x^7 + 15060*x^8 + 41382*x^9 + 340657*x^10 + 950061*x^11 + 8132676*x^12 + ...

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

%e A(x)^2 = 1 + 2*x + 7*x^2 + 20*x^3 + 103*x^4 + 328*x^5 + 1941*x^6 + 6506*x^7 + 41382*x^8 + 142892*x^9 + 950061*x^10 + ...

%e A(x)^3 = 1 + 3*x + 12*x^2 + 40*x^3 + 198*x^4 + 723*x^5 + 3927*x^6 + 15060*x^7 + 86190*x^8 + 340657*x^9 + 2016195*x^10 + ...

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

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

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

%e SPECIFIC VALUES.

%e The g.f. A(x) converges at the radius of convergence r, given by

%e A(-r) = 1 at r = (A(r) - 1)/(1 + A(r)^2) = 0.1795090246029167685576...

%e where A(r) = (1 + (28 + sqrt(783))^(1/3) + (28 - sqrt(783))^(1/3))/3 = 1.6956207695598620574163671... solves A(r)^3 - A(r)^2 = 2.

%e Other values are as follows.

%e A(t) = 3/2 at t = 0.1762576405478293392948378476047094214871919048852854...

%e with A(-t) = 0.9457634131178785046715685513829104426794138117773372...

%e A(1/6) = 1.39045291214794641706750008755820521981873579773148377...

%e A(-1/6) = 0.92547553450368274047514062093278734252641968691372863...

%e A(1/7) = 1.26282273990610251025800463852287012565418776197621997...

%e A(-1/7) = 0.91531855101291210815598364280272856428949318592006407...

%e A(1/8) = 1.20403758075277993770588254622742634950821058062345547...

%e A(-1/8) = 0.91758011120888933832570407861048171782335413914549218...

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

%o 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. A368629, A368626, A368628.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jan 09 2024