A340360 G.f. A(x) satisfies: A(x) = Sum_{n>=0} (n+1) * x^n / (1 - x^(n+1)*A(x)^n)^3.

1, 5, 9, 20, 50, 102, 365, 1279, 3894, 13925, 48903, 169524, 621128, 2227855, 7970244, 29200251, 107135216, 395895525, 1475549405, 5498774091, 20567632890, 77314123025, 291467136572, 1103022636621, 4187756743759, 15934627999334, 60781416980898, 232363965399736

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 = 2, and t = 3.

Links

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

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)^3.

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

Example

G.f.: A(x) = 1 + 5*x + 9*x^2 + 20*x^3 + 50*x^4 + 102*x^5 + 365*x^6 + 1279*x^7 + 3894*x^8 + 13925*x^9 + 48903*x^10 + ...
where
A(x) = 1/(1 - x)^3 + 2*x/(1 - x^2*A(x))^3 + 3*x^2/(1 - x^3*A(x)^2)^3 + 4*x^3/(1 - x^4*A(x)^3)^3 + 5*x^4/(1 - x^5*A(x)^4)^3 + ...
also
A(x) = 1/(1 - x)^2 + 3*x/(1 - x^2*A(x))^2 + 6*x^2/(1 - x^3*A(x)^2)^2 + 10*x^3/(1 - x^4*A(x)^3)^2 + 15*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))^3 )); 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, (m+1)*(m+2)/2 * 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, A340359.

Sequence in context: A011983 A087940 A092387 * A323110 A272786 A273150

Adjacent sequences: A340357 A340358 A340359 * A340361 A340362 A340363

Keyword

nonn

Author

Paul D. Hanna, Jan 07 2021

Status

approved