A340333 G.f. A(x) satisfies: A(x) = Sum_{n>=0} x^n/(1 - x^2*A(x)^n).

1, 1, 2, 2, 4, 7, 16, 38, 97, 257, 704, 1985, 5742, 17013, 51532, 159356, 502470, 1613880, 5275917, 17543426, 59307258, 203759160, 711246902, 2521876015, 9081377033, 33207738613, 123289411854, 464675856111, 1777656126662, 6901581699899, 27187917004378

Offset

0,3

Comments

The g.f. A(x) of this sequence is motivated by the following identity:

Sum_{n>=0} p^n/(1 - q*r^n) = Sum_{n>=0} q^n/(1 - p*r^n) = Sum_{n>=0} p^n*q^n*r^(n^2)*(1 - p*q*r^(2*n))/((1 - p*r^n)*(1 - q*r^n)) ;

here, p = x, q = x^2, and r = A(x).

Links

Table of n, a(n) for n=0..30.

Formula

G.f. A(x) satisfies:

(1) A(x) = Sum_{n>=0} x^n / (1 - x^2*A(x)^n).

(2) A(x) = Sum_{n>=0} x^(2*n) / (1 - x*A(x)^n).

(3) A(x) = Sum_{n>=0} x^(3*n) * A(x)^(n^2) * (1 - x^3*A(x)^(2*n)) / ((1 - x*A(x)^n)*(1 - x^2*A(x)^n)).

Example

G.f.: A(x) =  = 1 + x + 2*x^2 + 2*x^3 + 4*x^4 + 7*x^5 + 16*x^6 + 38*x^7 + 97*x^8 + 257*x^9 + 704*x^10 + 1985*x^11 + 5742*x^12 + ...
where
A(x) = 1/(1 - x^2) + x/(1 - x^2*A(x)) + x^2/(1 - x^2*A(x)^2) + x^3/(1 - x^2*A(x)^3) + x^4/(1 - x^2*A(x)^4) + ...
also
A(x) = 1/(1 - x) + x^2/(1 - x*A(x)) + x^4/(1 - x*A(x)^2) + x^6/(1 - x*A(x)^3) + x^8/(1 - x*A(x)^4) + ...

Prog

(PARI) {a(n) = my(A=1); for(i=1, n, A = sum(m=0, n, x^m /(1 - x^2*A^m +x*O(x^n)) )); polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n) = my(A=1); for(i=1, n, A = sum(m=0, n, x^(2*m) /(1 - x*A^m +x*O(x^n)) )); polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A340356.

Sequence in context: A014266 A032441 A238184 * A065844 A131199 A112059

Adjacent sequences: A340330 A340331 A340332 * A340334 A340335 A340336

Keyword

nonn

Author

Paul D. Hanna, Jan 11 2021

Status

approved