A368635 Expansion of g.f. A(x) satisfying A(x) = 1 + 3*x*A(x)^2 - 2*x*A(-x)^2.

1, 1, 10, 21, 310, 762, 12820, 33805, 607550, 1667214, 31182540, 87799362, 1686609820, 4835044372, 94676506920, 275037241149, 5463738069390, 16035014605830, 322140216214300, 953095126595062, 19320606147948820, 57539265876939756, 1175037853461723160, 3518503980453113106

Offset

0,3

Comments

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

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

(1.b) A(x) = 1 + x*(A(x)^2 + A(-x)^2)/2 + 5*x*(A(x)^2 - A(-x)^2)/2.

(2.a) (A(x) + A(-x))/2 = 1 + 5*x*(A(x)^2 - A(-x)^2)/2.

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

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

(3.a) A(x) = (1 - sqrt(1 - 40*x + 20*x*A(-x) + 100*x^2*A(-x)^2))/(10*x).

(3.b) A(-x) = (sqrt(1 + 40*x - 20*x*A(x) + 100*x^2*A(x)^2) - 1)/(10*x).

(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).

(5) 0 = (2 - x) - 2*(1-3*x)*A(x) - x*(3+10*x)*A(x)^2 + 30*x^2*A(x)^3 - 25*x^3*A(x)^4.

(6) x = (1 + 3*x*A(x) - 5*x^2*A(x)^2) - sqrt(1 + 4*x*A(x) - 4*x^2*A(x)^2).

(7) A(x) = (1/x)*Series_Reversion( (1 + 3*x - 5*x^2) - sqrt(1 + 4*x - 4*x^2) ).

(8.a) Sum_{n>=0} a(n) * (sqrt(6) - 2)^n/5^n = sqrt(6)/2.

(8.b) Sum_{n>=0} a(n) * (2 - sqrt(6))^n/5^n = 1.

D-finite with recurrence +10935*n*(n-1)*(75132*n-217883) *(n+1)*a(n) -1458*n*(n-1) *(865728*n^2 -6402143*n+11031849)*a(n-1) -360 *(n-1)*(159580368*n^3 -781944228*n^2 +1079436906*n -451430219)*a(n-2) +24*(3677612544*n^4 -41906753640*n^3 +172755991440*n^2 -308392913875*n +202512185406)*a(n-3) +12000*(2*n-7) *(300528*n^3 -1622852*n^2 +3317670*n -2937917)*a(n-4) -3200*(n-4)*(865728*n -2073503) *(2*n-7)*(2*n-9) *a(n-5)=0. - R. J. Mathar, Jan 25 2024

Example

G.f.: A(x) = 1 + x + 10*x^2 + 21*x^3 + 310*x^4 + 762*x^5 + 12820*x^6 + 33805*x^7 + 607550*x^8 + 1667214*x^9 + 31182540*x^10 + ...
RELATED SERIES.
We can see from the expansion of A(x)^2, which begins
A(x)^2 = 1 + 2*x + 21*x^2 + 62*x^3 + 762*x^4 + 2564*x^5 + 33805*x^6 + 121510*x^7 + 1667214*x^8 + 6236508*x^9 + 87799362*x^10 + ...
that the odd bisection of A(x) is derived from the even bisection of A(x)^2:
(A(x) - A(-x))/2 = x + 21*x^3 + 762*x^5 + 33805*x^7 + 1667214*x^9 + ...
(A(x)^2 + A(-x)^2)/2 = 1 + 21*x^2 + 762*x^4 + 33805*x^6 + 1667214*x^8 + ...
and the even bisection of A(x) is derived from the odd bisection of A(x)^2:
(A(x) + A(-x))/2 = 1 + 10*x^2 + 310*x^4 + 12820*x^6 + 607550*x^8 + 31182540*x^10 + ...
(A(x)^2 - A(-x)^2)/2 = 2*x + 62*x^3 + 2564*x^5 + 121510*x^7 + 6236508*x^9 + ...
so that (A(x) + A(-x))/2 = 1 + 5*x * (A(x)^2 - A(-x)^2)/2.
SPECIFIC VALUES.
A(-r) = 1 and A(r) = sqrt(6)/2 at r = (sqrt(6) - 2)/5 = 0.0898979485566356....

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 + 5*x*(A^2 - B^2)/2 ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = my(A); A = (1/x)*serreverse( (1 + 3*x - 5*x^2) - sqrt(1 + 4*x - 4*x^2 +x^2*O(x^n)) ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A368633, A368634, A368629, A368627.

Sequence in context: A153317 A110418 A336759 * A336748 A041198 A035318

Adjacent sequences: A368632 A368633 A368634 * A368636 A368637 A368638

Keyword

nonn

Author

Paul D. Hanna, Jan 12 2024

Status

approved