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))^4 * F(n)^2 for n > 0.
EXAMPLE
G.f.: A(x) = x + x^3 + 3*x^5 + 12*x^7 + 56*x^9 + 282*x^11 + 1494*x^13 + 8207*x^15 + 46332*x^17 + 267174*x^19 + ...
The series reversion is here denoted R(x) so that R(A(x)) = x where
R(x) = x - x^3 - x^9 + 3*x^11 - 3*x^13 + x^15 - x^33 + 9*x^35 - 36*x^37 + 84*x^39 + ... + A350483(n)*x^(2*n-1) + ...
and which by definition also satisfies A(x^3*R(x)) = x^4 - x^6.
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*6^(n-1), by the following recurrence:
F(0) = x,
F(1) = (1 - x^2),
F(2) = (1 - x^8 * (1-x^2)^2),
F(3) = (1 - x^32 * (1-x^2)^8 * F(2)^2),
F(4) = (1 - x^128 * (1-x^2)^32 * F(2)^8 * F(3)^2),
F(5) = (1 - x^512 * (1-x^2)^128 * F(2)^32 * F(3)^8 * F(4)^2),
...
F(n+1) = 1 - (1 - F(n))^4 * 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^8*(1-x^2)^2) * (1 - x^32*(1-x^2)^8*(1 - x^8*(1-x^2)^2)^2) * (1 - x^128*(1-x^2)^32*(1 - x^8*(1-x^2)^2)^8*(1 - x^32*(1-x^2)^8*(1 - x^8*(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^4*(1 - x^2) - subst(x*Ser(A), x, x^3 * serreverse(x*Ser(A))), #A+3) ); A[n]}
for(n=1, 30, print1(a(2*n-1), ", "))
(PARI) /* Using Infinite Product Formula for Series Reversion */
N = 300; \\ set limit on order of polynomials to be 2 times desired number of terms
{F(n) = my(G=x); if(n==0, G=x, if(n==1, G = (1-x^2), G = 1 - (1 - F(n-1))^4 * F(n-1)^2 +x*O(x^N) )); G}
{a(n) = my(A = prod(k=0, #binary(n), F(k) +x*O(x^n))); polcoeff(serreverse(A), n)}
for(n=1, 30, print1(a(2*n-1), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 01 2022
STATUS
approved