A340361 G.f. A(x) satisfies: A(x) = (1-x) * Sum_{n>=0} x^n / (1 - x*A(x)^n).

1, 1, 1, 2, 5, 14, 42, 133, 440, 1510, 5347, 19459, 72561, 276616, 1076236, 4268236, 17238623, 70858091, 296293158, 1260044245, 5449129205, 23962691920, 107160352895, 487379459886, 2254710459801, 10611155135759, 50808249311687, 247538711398811

Offset

0,4

Comments

The g.f. 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, and r = A(x).

Links

Table of n, a(n) for n=0..27.

Formula

G.f. A(x) satisfies:

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

(2) A(x) = (1-x) * Sum_{n>=0} x^(2*n) * A(x)^(n^2) * (1 + x*A(x)^n) / (1 - x*A(x)^n). - Paul D. Hanna, Feb 20 2021

Example

G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 + 42*x^6 + 133*x^7 + 440*x^8 + 1510*x^9 + 5347*x^10 + 19459*x^11 + 72561*x^12 + ...
where
A(x)/(1-x) = 1/(1-x) + x/(1 - x*A(x)) + x^2/(1 - x*A(x)^2) + x^3/(1 - x*A(x)^3) + x^4/(1 - x*A(x)^4) + x^5/(1 - x*A(x)^5) + ...

Prog

(PARI) {a(n) = my(A=1); for(i=1, n, A = (1-x) * sum(m=0, n, x^m / (1 - x*A^m +x*O(x^n)) ) ); polcoeff(H=A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A340891.

Sequence in context: A126569 A162748 A061815 * A308329 A202061 A346660

Adjacent sequences: A340358 A340359 A340360 * A340362 A340363 A340364

Keyword

nonn

Author

Paul D. Hanna, Jan 25 2021

Status

approved