OFFSET
1,3
FORMULA
G.f.: A(x) = Series_Reversion( Product_{n>=0} F(n) ), where F(0) = x, F(1) = 1-x^3, and F(n+1) = 1 - (1 - F(n))^2 * F(n)^3 for n > 0.
EXAMPLE
G.f.: A(x) = x + x^4 + 5*x^7 + 29*x^10 + 203*x^13 + 1519*x^16 + 12047*x^19 + 99112*x^22 + 839142*x^25 + ...
The series reversion is here denoted R(x) so that R(A(x)) = x where
R(x) = x - x^4 - x^7 + 4*x^10 - 7*x^13 + 11*x^16 - 18*x^19 - 5*x^22 + 138*x^25 - 368*x^28 + ... + A350477(n)*x^(3*n-2) + ...
and which by definition also satisfies A(x*R(x)) = x^2 - x^5.
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 3*5^(n-1), by the following recurrence:
F(0) = x,
F(1) = (1 - x^3),
F(2) = (1 - x^6 * (1-x^3)^3),
F(3) = (1 - x^12 * (1-x^3)^6 * F(2)^3),
F(4) = (1 - x^24 * (1-x^3)^12 * F(2)^6 * F(3)^3),
F(5) = (1 - x^48 * (1-x^3)^24 * F(2)^12 * F(3)^6 * F(4)^3),
...
F(n+1) = 1 - (1 - F(n))^2 * F(n)^3
...
Then the g.f. A(x) equals the infinite product:
R(x) = x * F(1) * F(2) * F(3) * ... * F(n) * ...
that is,
R(x) = x * (1-x^3) * (1 - x^6*(1-x^3)^3) * (1 - x^12*(1-x^3)^6*(1 - x^6*(1-x^3)^3)^3) * (1 - x^24*(1-x^3)^12*(1 - x^6*(1-x^3)^3)^6*(1 - x^12*(1-x^3)^6*(1 - x^6*(1-x^3)^3)^3)^3) * ...
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^3) - subst(x*Ser(A), x, x * serreverse(x*Ser(A))), #A+1) ); A[n]}
for(n=1, 30, print1(a(3*n-2), ", "))
(PARI) /* Using Infinite Product Formula for Series Reversion */
N = 300; \\ set limit on order of polynomials to be 3 times desired number of terms
{F(n) = my(G=x); if(n==0, G=x, if(n==1, G = (1-x^3), G = 1 - (1 - F(n-1))^2 * F(n-1)^3 +x^2*O(x^N) )); G}
{a(n) = my(A, B = prod(k=0, #binary(n), F(k) +x*O(x^n)));
A = serreverse(B); polcoeff(H=A, n)}
for(n=1, 30, print1(a(3*n-2), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 01 2022
STATUS
approved