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
Keywords
Examples
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.
Links
- Paul D. Hanna, Table of n, a(n) for n = 0..400
Programs
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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),", "))
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
Comments