A357792 a(n) = coefficient of x^n in A(x) = Sum_{n>=0} C(x)^n * (1 - C(x)^n)^n, where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).

1, 1, 1, 3, 7, 20, 60, 189, 613, 2039, 6918, 23850, 83315, 294282, 1049279, 3771685, 13653313, 49730599, 182130129, 670274170, 2477514172, 9193599339, 34237330355, 127914531260, 479318575375, 1800971051420, 6783809423496, 25611913597250, 96903193235645, 367363376407250

Offset

0,4

Comments

Related identity: 0 = Sum_{n=-oo..+oo} x^n * (1 - x^n)^n, which holds as a formal power series in x.

Related identity: 0 = Sum_{n=-oo..+oo} x^n * (1 - C(x)^n)^n / (1 - C(x))^n, where C(x) = x + C(x)^2.

Links

Paul D. Hanna, Table of n, a(n) for n = 0..400

Formula

Given C(x) = x + C(x)^2, g.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by:

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

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

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

(4) A(x) = Sum_{n>=1} -(-1/x)^n * C(x)^(n^2) / [ (1 - C(x)^n) / (1 - C(x)) ]^n.

a(n) ~ c * 2^(2*n) / n^(3/2), where c = 0.1930490961334149255878338532701052858837... - Vaclav Kotesovec, Mar 14 2023

Example

G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 7*x^4 + 20*x^5 + 60*x^6 + 189*x^7 + 613*x^8 + 2039*x^9 + 6918*x^10 + 23850*x^11 + 83315*x^12 + ...
Let C = C(x) = x + C(x)^2, then
A(x) = 1 + C*(1 - C) + C^2*(1 - C^2)^2 + C^3*(1 - C^3)^3 + C^4*(1 - C(x)^4)^4 + C^5*(1 - C(x)^5)^5 + ... + C(x)^n * (1 - C(x)^n)^n + ...
also,
A(x) = 1 + x*(1) + x^2*(1 + C)^2 + x^3*(1 + C + C^2)^3 + x^4*(1 + C + C^2 + C^3)^4 + x^5*(1 + C + C^2 + C^3 + C^4)^5 + x^6*(1 + C + C^2 + C^3 + C^4 + C^5)^6 + ... + x^n*(1 + C + C^2 + C^3 + ... + C^(n-1))^n + ...
further,
A(x) = 1/(1 - C) - C^2/(1 - C^2)^2 + C^6/(1 - C^3)^3 - C^12/(1 - C^4)^4 + C^20/(1 - C^5)^5 + ... + (-1)^(n-1) * C(x)^(n*(n-1)) / (1 - C^n)^n + ...
where the related Catalan series, C(x) = (1 - sqrt(1 - 4*x))/2, begins:
C(x) = x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 + 42*x^6 + 132*x^7 + 429*x^8 + 1430*x^9 + 4862*x^10 + 16796*x^11 + 58786*x^12 + ... + A000108(n)*x^(n+1) + ...
SPECIFIC VALUES.
The radius of convergence of the power series A(x) equals 1/4.
The power series A(x) converges at x = 1/4 to
A(1/4) = 1.578564238051657388445969550353857020762848420638921268996...
which equals the following sums:
(1) A(1/4) = Sum_{n>=0} (2^n - 1)^n / 2^(n*(n+1)),
(2) A(1/4) = Sum_{n>=1} (-1)^(n-1) * 2^n / (2^n - 1)^n.

Prog

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

Crossrefs

Cf. A357793, A000108.

Sequence in context: A132364 A110490 A132868 * A361625 A056783 A320735

Adjacent sequences: A357789 A357790 A357791 * A357793 A357794 A357795

Keyword

nonn

Author

Paul D. Hanna, Dec 14 2022

Status

approved