A350475 G.f. A(x) satisfies: A(x) = A(x^2 - x^4)/x.

1, -1, -1, 3, -4, 6, -7, -11, 54, -68, -33, 207, -160, -398, 1087, -461, -3370, 11120, -18761, 9287, 52861, -191783, 325226, -164758, -710654, 2437586, -4100514, 2865198, 6070531, -26403315, 51585667, -50596137, -38932192, 280507198

Offset

1,4

Links

Table of n, a(n) for n=1..34.

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^(2*n-1) satisfies:

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

(2) R(x*A(x)) = x^2 - x^4, where R(A(x)) = x.

(3) A(x) = Product_{n>=0} F(n), where F(0) = x, F(1) = 1-x^2, and F(n+1) = 1 - (1 - F(n))^2 * F(n)^2 for n > 0.

Example

G.f.: A(x) = x - x^3 - x^5 + 3*x^7 - 4*x^9 + 6*x^11 - 7*x^13 - 11*x^15 + 54*x^17 - 68*x^19 - 33*x^21 + 207*x^23 - 160*x^25 + ...
The series reversion is here denoted R(x) so that R(A(x)) = x where
R(x) = x + x^3 + 4*x^5 + 17*x^7 + 89*x^9 + 487*x^11 + 2835*x^13 + 17039*x^15 + 105390*x^17 + ... + A350474(n)*x^(2*n-1) + ...
and which by definition also satisfies R(x*A(x)) = x^2 - x^4.
GENERATING METHOD.
One may generate the g.f. A(x) using the following method.
Define F(n), a polynomial in x of order 2^(2*n-1), by the following recurrence:
F(0) = x,
F(1) = (1 - x^2),
F(2) = (1 - x^4 * (1-x^2)^2),
F(3) = (1 - x^8 * (1-x^2)^4 * F(2)^2),
F(4) = (1 - x^16 * (1-x^2)^8 * F(2)^4 * F(3)^2),
F(5) = (1 - x^32 * (1-x^2)^16 * F(2)^8 * F(3)^4 * F(4)^2),
...
F(n+1) = 1 - (1 - F(n))^2 * F(n)^2
...
Then the g.f. A(x) equals the infinite product:
A(x) = x * F(1) * F(2) * F(3) * ... * F(n) * ...
that is,
A(x) = x * (1-x^2) * (1 - x^4*(1-x^2)^2) * (1 - x^8*(1-x^2)^4*(1 - x^2*(1-x^2)^2)^2) * (1 - x^16*(1-x^2)^8*(1 - x^2*(1-x^2)^2)^4*(1 - x^4*(1-x^2)^4*(1 - x^2*(1-x^2)^2)^2)^2) * ...
SPECIFIC VALUES.
The infinite product formula allows us to evaluate the function A(x) at certain x rather quickly.
A(1/2) = (1/2) * (3/2^2) * (247/2^8) * (4290025567/2^32) * ... = 0.36139962803961485982912810261...
A(2/3) = (2/3) * (5/3^2) * (6161/3^8) * (1846946921491841/3^32) * ... = 0.34664667253812499241960389380...
A(1/3) = (1/3) * (8/3^2) * (6497/3^8) * (1852847292558977/3^32) * ... = 0.29337866246734888661663159512...
The first relative maximum value of A(x) is given by
A(0.554161678235...) = 0.365908834442...

Prog

(PARI) {a(n) = my(A, R=[1, 0]); for(i=1, n, R=concat(R, 0);
R[#R] = -polcoeff( x^2*(1 - x^2) - subst(x*Ser(R), x, x * serreverse(x*Ser(R))), #R+1) );
A=Vec(serreverse(x*Ser(R))); A[n]}
for(n=1, 40, print1(a(2*n-1), ", "))
(PARI) /* Using Infinite Product Formula for Series Reversion */
{F(n) = my(G=x); if(n==0, G=x, if(n==1, G = (1-x^2), G = 1 - (1 - F(n-1))^2 * F(n-1)^2 )); G}
{a(n) = my(A = prod(k=0, #binary(n), F(k) +x*O(x^n))); polcoeff(A, n)}
for(n=1, 40, print1(a(2*n-1), ", "))

Crossrefs

Cf. A350474 (inverse), A273218, A350431, A350433, A350477, A350479, A350481, A350483.

Sequence in context: A073906 A108797 A089161 * A060965 A153883 A033162

Adjacent sequences: A350472 A350473 A350474 * A350476 A350477 A350478

Keyword

sign

Author

Paul D. Hanna, Jan 01 2022

Status

approved