A340359 G.f. A(x) satisfies: A(x) = Sum_{n>=0} (n+1) * x^n / (1 - x^(n+1)*A(x)^n).

1, 3, 4, 7, 12, 20, 52, 124, 297, 802, 2114, 5705, 15653, 42392, 116560, 324503, 907520, 2556402, 7223284, 20471723, 58319247, 166859181, 479305506, 1381683897, 3993923929, 11574493329, 33625052782, 97908062011, 285724318094, 835602499442

Offset

0,2

Comments

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

Sum_{n>=0} C(t+n-1,n) * p^n/(1 - q*r^n)^s = Sum_{n>=0} C(s+n-1,n) * q^n/(1 - p*r^n)^t ;

here, p = x, q = x, r = x*A(x), s = 1, and t = 2.

Links

Table of n, a(n) for n=0..29.

Formula

G.f. A(x) satisfies the following relations.

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

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

Example

G.f.: A(x) = 1 + 3*x + 4*x^2 + 7*x^3 + 12*x^4 + 20*x^5 + 52*x^6 + 124*x^7 + 297*x^8 + 802*x^9 + 2114*x^10 + 5705*x^11 + 15653*x^12 + ...
where
A(x) = 1/(1 - x) + 2*x/(1 - x^2*A(x)) + 3*x^2/(1 - x^3*A(x)^2) + 4*x^3/(1 - x^4*A(x)^3) + 5*x^4/(1 - x^5*A(x)^4) + ...
also
A(x) = 1/(1 - x)^2 + x/(1 - x^2*A(x))^2 + x^2/(1 - x^3*A(x)^2)^2 + x^3/(1 - x^4*A(x)^3)^2 + x^4/(1 - x^5*A(x)^4)^2 + ...

Prog

(PARI) {a(n) = my(A=1); for(i=1, n, A = sum(m=0, n, (m+1) * x^m / (1 - x^(m+1)*A^m +x*O(x^n)) )); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = my(A=1); for(i=1, n, A = sum(m=0, n, x^m / (1 - x^(m+1)*A^m +x*O(x^n))^2 )); ; polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A340329, A340338, A340355, A340356, A340357, A340358, A340360.

Sequence in context: A050342 A293642 A214286 * A108700 A325851 A062202

Adjacent sequences: A340356 A340357 A340358 * A340360 A340361 A340362

Keyword

nonn

Author

Paul D. Hanna, Jan 07 2021

Status

approved