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))^3 * F(n)^3 for n > 0.
EXAMPLE
G.f.: A(x) = x + x^4 + 4*x^7 + 23*x^10 + 150*x^13 + 1060*x^16 + 7910*x^19 + 61319*x^22 + 488982*x^25 + ...
The series reversion is here denoted R(x) so that R(A(x)) = x where
R(x) = x - x^4 - x^10 + 4*x^13 - 6*x^16 + 4*x^19 - x^22 - x^28 + 10*x^31 - 45*x^34 + ... + A350481(n)*x^(3*n-2) + ...
and which by definition also satisfies A(x^2*R(x)) = x^3 - 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 3*6^(n-1), by the following recurrence:
F(0) = x,
F(1) = (1 - x^3),
F(2) = (1 - x^9 * (1-x^3)^3),
F(3) = (1 - x^27 * (1-x^3)^9 * F(2)^3),
F(4) = (1 - x^81 * (1-x^3)^27 * F(2)^9 * F(3)^3),
F(5) = (1 - x^243 * (1-x^3)^81 * F(2)^27 * F(3)^9 * F(4)^3),
...
F(n+1) = 1 - (1 - F(n))^3 * F(n)^3
...
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^3) * (1 - x^9*(1-x^3)^3) * (1 - x^27*(1-x^3)^9*(1 - x^9*(1-x^3)^3)^3) * (1 - x^81*(1-x^3)^27*(1 - x^9*(1-x^3)^3)^9*(1 - x^27*(1-x^3)^9*(1 - x^9*(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^3*(1 - x^3) - subst(x*Ser(A), x, x^2 * serreverse(x*Ser(A))), #A+2) ); 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 2 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))^3 * F(n-1)^3 +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(3*n-2), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 01 2022
STATUS
approved