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A357792
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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).
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2
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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
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OFFSET
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0,4
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COMMENTS
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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.
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LINKS
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FORMULA
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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
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EXAMPLE
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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.
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PROG
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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), ", "))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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