A350432 G.f. A(x) satisfies: A(x) = A(x^2 + x^3) / x.

1, 1, 1, 2, 2, 3, 5, 9, 14, 18, 25, 38, 60, 97, 159, 265, 444, 735, 1187, 1865, 2851, 4271, 6378, 9621, 14724, 22864, 35947, 57044, 91141, 146384, 236102, 382124, 620298, 1009685, 1647703, 2694709, 4413524, 7232548, 11845740, 19369888, 31590755, 51346902, 83126317

Offset

1,4

Comments

The radius of convergence r of the g.f. A(x) is r = (sqrt(5) - 1)/2. - Paul D. Hanna, Jan 26 2024

Links

Paul D. Hanna, Table of n, a(n) for n = 1..2050

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies:

(1) A(x) = A(x^2 + x^3) / x.

(2) R(x*A(x)) = x^2 + x^3, where R(A(x)) = x.

(3) A(x) = x * Product_{n>=1} F(n), where F(1) = 1+x, and F(n+1) = 1 + (F(n) - 1)^2 * F(n) for n >= 1.

Example

G.f.: A(x) = x + x^2 + x^3 + 2*x^4 + 2*x^5 + 3*x^6 + 5*x^7 + 9*x^8 + 14*x^9 + 18*x^10 + 25*x^11 + 38*x^12 + 60*x^13 + ...
Let R(x) be the series reversion of A(x),
R(x) = x - x^2 + x^3 - 2*x^4 + 6*x^5 - 17*x^6 + 45*x^7 - 123*x^8 + 360*x^9 - 1085*x^10 + 3271*x^11 - 9905*x^12 + 30417*x^13 + ... + A350431(n)*x^n + ...
then R(x) and g.f. A(x) satisfy:
(1) R(A(x)) = x,
(2) R(x*A(x)) = x^2 + x^3.
GENERATING METHOD.
Define F(n), a polynomial in x of order 3^(n-1), by the following recurrence:
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 + (F(n) - 1)^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)))) * ...

Prog

(PARI) /* Using Functional Equation in Definition */
{a(n) = my(A=x); for(i=0, #binary(n),
A = subst(A, x, x^2*(1 + x) +x^2*O(x^n) )/x ); polcoeff(A, n)}
for(n=1, 50, 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 + (F(n-1) - 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), ", "))

Crossrefs

Cf. A350431 (inverse), A273162, A350433, A350434.

Cf. A369545, A369546, A369547, A369548, A369549.

Cf. A369552, A369553, A369554, A369555, A369556.

Sequence in context: A348850 A183559 A080553 * A226956 A141602 A153931

Adjacent sequences: A350429 A350430 A350431 * A350433 A350434 A350435

Keyword

nonn

Author

Paul D. Hanna, Dec 30 2021

Status

approved