A369083 Expansion of g.f. A(x) satisfying A(x) = 1 + x*(5*A(x)^2 - A(-x)^2)/4.

1, 1, 3, 7, 30, 83, 402, 1199, 6180, 19232, 102939, 329217, 1807344, 5891442, 32936724, 108884607, 617125788, 2062285676, 11813994060, 39818644316, 230067933810, 780838528379, 4543410985386, 15509003672617, 90771938228244, 311354249554852, 1831389290870538, 6307784087296006

Offset

0,3

Comments

Conjecture: a(n) == binomial(4*n+3,n) (mod 2) for n >= 0 (cf. A263133).

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 + x*(5*A(x)^2 - A(-x)^2)/4.

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

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

(3.a) A(x) = (1 - sqrt(1 - 12*x + 6*x*A(-x) + 9*x^2*A(-x)^2)) / (3*x).

(3.b) A(-x) = (sqrt(1 + 12*x - 6*x*A(x) + 9*x^2*A(x)^2) - 1) / (3*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 = (1-4*x) - (1-10*x)*A(x) - (5+12*x)*x*A(x)^2 + 15*x^2*A(x)^3 - 9*x^3*A(x)^4.

(6) x = (1 + 10*x*A(x) - 12*x^2*A(x)^2 - sqrt(1 + 4*x*A(x) - 4*x^2*A(x)^2))/8.

(7) A(x) = (1/x)*Series_Reversion( (1 + 10*x - 12*x^2 - sqrt(1 + 4*x - 4*x^2))/8 ).

Example

G.f.: A(x) = 1 + x + 3*x^2 + 7*x^3 + 30*x^4 + 83*x^5 + 402*x^6 + 1199*x^7 + 6180*x^8 + 19232*x^9 + 102939*x^10 + ...
RELATED SERIES.
We can see from the expansion of A(x)^2, which begins
A(x)^2 = 1 + 2*x + 7*x^2 + 20*x^3 + 83*x^4 + 268*x^5 + 1199*x^6 + 4120*x^7 + 19232*x^8 + 68626*x^9 + 329217*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 + 7*x^3 + 83*x^5 + 1199*x^7 + 19232*x^9 + ...
(A(x)^2 + A(-x)^2)/2 = 1 + 7*x^2 + 83*x^4 + 1199*x^6 + 19232*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 + 3*x^2 + 30*x^4 + 402*x^6 + 6180*x^8 + 102939*x^10 + ...
(A(x)^2 - A(-x)^2)/2 = 2*x + 20*x^3 + 268*x^5 + 4120*x^7 + 68626*x^9 + ...
so that (A(x) + A(-x))/2 = 1 + (3/2)*x * (A(x)^2 - A(-x)^2)/2.

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 + (3/2)*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 + 10*x - 12*x^2 - sqrt(1 + 4*x - 4*x^2  +x^2*O(x^n) ))/8 ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A368633, A368634, A368635, A368627, A368629.

Sequence in context: A333393 A227077 A136934 * A201794 A188229 A277739

Adjacent sequences: A369080 A369081 A369082 * A369084 A369085 A369086

Keyword

nonn

Author

Paul D. Hanna, Jan 12 2024

Status

approved