A368627 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.

1, 1, 3, 7, 40, 103, 723, 1941, 15060, 41382, 340657, 950061, 8132676, 22916139, 201684153, 572618987, 5145063940, 14692661910, 134152006842, 384852888898, 3559210821120, 10248531332559, 95777105998365, 276630878235275, 2607824127882204, 7551545042631558, 71714198513326425

Offset

0,3

Links

Paul D. Hanna, Table of n, a(n) for n = 0..600

Formula

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

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

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

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

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

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

Example

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 + ...
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
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 + ...
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 + ...
so that the bisections of the above series are related by
(A(x) - A(-x))/2 = x*(A(x)^2 + A(-x)^2)/2, and
(A(x) + A(-x))/2 = 1 + x*(A(x)^3 - A(-x)^3)/2.
SPECIFIC VALUES.
The g.f. A(x) converges at the radius of convergence r, given by
A(-r) = 1 at r = (A(r) - 1)/(1 + A(r)^2) = 0.1795090246029167685576...
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.
Other values are as follows.
A(t) = 3/2 at t = 0.1762576405478293392948378476047094214871919048852854...
with A(-t) = 0.9457634131178785046715685513829104426794138117773372...
A(1/6) = 1.39045291214794641706750008755820521981873579773148377...
A(-1/6) = 0.92547553450368274047514062093278734252641968691372863...
A(1/7) = 1.26282273990610251025800463852287012565418776197621997...
A(-1/7) = 0.91531855101291210815598364280272856428949318592006407...
A(1/8) = 1.20403758075277993770588254622742634950821058062345547...
A(-1/8) = 0.91758011120888933832570407861048171782335413914549218...

Prog

(PARI) {a(n) = my(A=1+x, B); 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)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A368629, A368626, A368628.

Sequence in context: A105621 A181081 A166895 * A018998 A018941 A018967

Adjacent sequences: A368624 A368625 A368626 * A368628 A368629 A368630

Keyword

nonn

Author

Paul D. Hanna, Jan 09 2024

Status

approved