OFFSET
1,4
FORMULA
G.f. A(x) = Sum_{n>=1} a(n)*x^(3*n-2) satisfies:
(1) A(x) = A(x^2 - x^5)/x.
(2) R(x*A(x)) = x^2 - x^5, where R(A(x)) = x.
(3) A(x) = 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 - x^7 + 4*x^10 - 7*x^13 + 11*x^16 - 18*x^19 - 5*x^22 + 138*x^25 - 368*x^28 + 298*x^31 + 819*x^34 + ...
The series reversion is here denoted R(x) so that R(A(x)) = x where
R(x) = x + x^4 + 5*x^7 + 29*x^10 + 203*x^13 + 1519*x^16 + 12047*x^19 + 99112*x^22 + ... + A350476(n)*x^(3*n-2) + ...
and which by definition also satisfies R(x*A(x)) = x^2 - x^5.
GENERATING METHOD.
One may generate the g.f. A(x) using the following method.
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:
A(x) = x * F(1) * F(2) * F(3) * ... * F(n) * ...
that is,
A(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) * ...
SPECIFIC VALUES.
The infinite product formula allows us to evaluate the function A(x) at certain x rather quickly.
A(1/2) = (1/2) * (7/2^3) * (32425/2^15) * (37774921088925702693943/2^75) * ... = 0.43287449035613254837986255178...
A(2/3) = (2/3) * (19/3^3) * (13909931/3^15) * (607748159091514279939881665134154491/3^75) * ... = 0.45439544136368649551815752865...
A(1/3) = (1/3) * (26/3^3) * (14331331/3^15) * (608265878429109169424505294701548091/3^75) * ... = 0.32059399676030866078416260973...
The first relative maximum value of A(x) is given by
A(0.6158833582739456...) = 0.46015216135279854...
PROG
(PARI) {a(n) = my(A, R=[1, 0]); for(i=1, n, R=concat(R, 0);
R[#R] = -polcoeff( x^2*(1 - x^3) - 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(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 = prod(k=0, #binary(n), F(k) +x*O(x^n))); polcoeff(A, n)}
for(n=1, 40, print1(a(3*n-2), ", "))
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Jan 01 2022
STATUS
approved