A273218 G.f. A(x) satisfies: A(x) = A(x^2 - x^3)/x.

1, -1, -1, 2, -2, 3, -1, -7, 10, 2, -15, 2, 34, -51, 17, 73, -218, 323, -135, -467, 1139, -1279, 430, 1587, -4274, 5798, -3249, -5664, 19061, -26208, 9976, 34430, -77516, 62473, 45193, -186383, 173814, 186306, -747220, 754744, 678009, -3221980, 4339761, -491173, -8988984, 17693649, -12889827, -13658278, 48487713

Offset

1,4

Comments

Equals the series reversion of the g.f. of A273162.

Links

Paul D. Hanna, Table of n, a(n) for n = 1..1030

Formula

G.f.: A(x) = Product_{n>=0} F(n), where F(0) = x, F(1) = 1-x, and F(n+1) = 1 - (1 - F(n))^2 * F(n) for n > 0. - Paul D. Hanna, Dec 30 2021

Example

G.f.: A(x) = x - x^2 - x^3 + 2*x^4 - 2*x^5 + 3*x^6 - x^7 - 7*x^8 + 10*x^9 + 2*x^10 - 15*x^11 + 2*x^12 + 34*x^13 - 51*x^14 + 17*x^15 + 73*x^16 - 218*x^17 +...
such that A(x) = A(x^2 - x^3)/x.
RELATED SERIES.
Let B(x) be the series reversion of g.f. A(x), so that B(A(x)) = x, then
B(x) = x + x^2 + 3*x^3 + 8*x^4 + 28*x^5 + 95*x^6 + 351*x^7 + 1309*x^8 + 5056*x^9 + 19787*x^10 + 78847*x^11 +...+ A273162(n)*x^n +...
such that B(x*A(x)) = x^2 - x^3.
From Paul D. Hanna, Dec 30 2021: (Start)
GENERATING METHOD.
Define F(n), a polynomial in x of order 3^(n-1), by the following recurrence:
F(0) = x,
F(1) = (1 - x),
F(2) = (1 - x^2 * (1-x)),
F(3) = (1 - x^4 * (1-x)^2 * F(2)),
F(4) = (1 - x^8 * (1-x)^4 * F(2)^2 * F(3)),
F(5) = (1 - x^16 * (1-x)^8 * F(2)^4 * F(3)^2 * F(4)),
...
F(n+1) = 1 - (1 - F(n))^2 * F(n)
...
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) * (1 - x^2*(1-x)) * (1 - x^4*(1-x)^2*(1 - x^2*(1-x))) * (1 - x^8*(1-x)^4*(1 - x^2*(1-x))^2*(1 - x^4*(1-x)^2*(1 - x^2*(1-x)))) * ...
(End)
From Paul D. Hanna, Dec 31 2021: (Start)
SPECIFIC VALUES.
The infinite product formula allows us to evaluate the function A(x) at certain x rather quickly.
A(1/2) = (1/2) * (1/2) * (7/2^3) * (505/2^9) * (134192983/2^27) * (2417851557060878608942777/2^81) * ... = 0.21571949163622469813172568...
A(2/3) = (2/3) * (1/3) * (23/3^3) * (19315/3^9) * (7622981770427/3^27) * ... = 0.18569744603728983530046038...
A(1/3) = (1/3) * (2/3) * (25/3^3) * (19583/3^9) * (7625401654987/3^27) * ... = 0.20471068371640502928595863...
The first relative maximum value of A(x) is given by
A(0.45249959935125940...) = 0.21748935249823157...
(End)

Prog

(PARI) {a(n) = my(A=x); for(i=1, #binary(n)+1, A = subst(A, x, x^2 - x^3 + x^2*O(x^n))/x); polcoeff(A, n) }
for(n=1, 60, print1(a(n), ", "))
(PARI) /* Using Infinite Product Formula */
{F(n) = my(G=x); if(n==0, G=x, if(n==1, G=1-x, G = 1 - (1 - F(n-1))^2*F(n-1) )); G}
{a(n) = my(A = prod(k=0, #binary(n), F(k) +x*O(x^n))); polcoeff(A, n)}
for(n=1, 50, print1(a(n), ", ")) \\ Paul D. Hanna, Dec 30 2021

Crossrefs

Cf. A273162, A350432.

Sequence in context: A016539 A345886 A220349 * A173160 A022461 A306821

Adjacent sequences: A273215 A273216 A273217 * A273219 A273220 A273221

Keyword

sign

Author

Paul D. Hanna, May 17 2016

Status

approved