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A120010
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G.f.: A(x) = (1-sqrt(1-4*x))/2 o x/(1-x) o (x-x^2), a composition of functions involving the Catalan function and its inverse.
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9
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1, 1, 1, 2, 6, 18, 53, 158, 481, 1491, 4688, 14913, 47913, 155261, 506881, 1665643, 5504988, 18287338, 61027991, 204499397, 687808931, 2321177071, 7857504876, 26673769002, 90783820081, 309720079813, 1058984020333, 3628267267358
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OFFSET
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1,4
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COMMENTS
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The n-th iteration of g.f. A(x) is: (1-sqrt(1-4*x))/2 o x/(1-n*x) o (x-x^2) = (1 - sqrt(1 - 4*(x-x^2)/(1-n*x+n*x^2) ))/2. See A120009 for the transpose of the composition of the same functions.
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LINKS
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FORMULA
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G.f.: A(x) = (1 - sqrt(1 - 4*(x-x^2)/(1-x+x^2) ))/2.
a(n)=sum{k=0..n, sum{j=0..n, (-1)^(n-j)*C(j+1,n-j)*C(j,k)*if(k<=j, A000108(j-k),0)}}. [offset 0]. [Paul Barry, Jan 28 2009]
Conjecture: n*a(n) +2*(4-3*n)*a(n-1) +(11*n-26)*a(n-2) +10*(3-n)*a(n-3) +5*(n-4)*a(n-4)= 0. - R. J. Mathar, Nov 14 2011
a(n) ~ sqrt(5*sqrt(5)-5) * (5+sqrt(5))^n / (sqrt(Pi) * n^(3/2) * 2^(n+7/2)). - Vaclav Kotesovec, Feb 13 2014
Equivalently, a(n) ~ 5^((n+1)/2) * phi^(n - 1/2) / (8 * sqrt(Pi) * n^(3/2)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 08 2021
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EXAMPLE
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G.f.: A(x) = x + x^2 + x^3 + 2*x^4 + 6*x^5 + 18*x^6 + 53*x^7 + 158*x^8 +...
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MATHEMATICA
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Rest[CoefficientList[Series[(1-Sqrt[1-4*(x-x^2)/(1-x+x^2)])/2, {x, 0, 20}], x]] (* Vaclav Kotesovec, Feb 13 2014 *)
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PROG
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(PARI) {a(n)=polcoeff((1 - sqrt(1 - 4*(x-x^2)/(1-x+x^2+x*O(x^n)) ))/2, n)}
for(n=1, 35, 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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