OFFSET
1,3
FORMULA
G.f.: A(x) = Series_Reversion( 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 + 4*x^5 + 17*x^7 + 89*x^9 + 487*x^11 + 2835*x^13 + 17039*x^15 + 105390*x^17 + 665364*x^19 + ...
The series reversion is here denoted R(x) so that A(R(x)) = x where
R(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 + ... + A350475(n)*x^(2*n-1) + ...
and which by definition also satisfies A(x*R(x)) = x^2 - x^4.
GENERATING METHOD.
One may obtain the g.f. A(x) from the following method used to generate the series reversion R(x).
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 series reversion R(x) equals the infinite product:
R(x) = x * F(1) * F(2) * F(3) * ... * F(n) * ...
that is,
R(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) * ...
The g.f. of this sequence is then obtained as the series reversion of this infinite product.
PROG
(PARI) {a(n) = my(A=[1, 0]); for(i=1, n, A=concat(A, 0);
A[#A] = -polcoeff( x^2*(1 - x^2) - subst(x*Ser(A), x, x * serreverse(x*Ser(A))), #A+1) ); A[n]}
for(n=1, 30, 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, B = prod(k=0, #binary(n), F(k) +x*O(x^n)));
A = serreverse(B); polcoeff(A, n)}
for(n=1, 30, print1(a(2*n-1), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 01 2022
STATUS
approved