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A214377
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G.f. satisfies: A(x) = 1 + 4*x*A(x)^(3/2).
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8
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1, 4, 24, 168, 1280, 10296, 86016, 739024, 6488064, 57946200, 524812288, 4808643120, 44493176832, 415146189360, 3901709352960, 36902658748320, 350980432461824, 3354743017001880, 32207616155320320, 310446853795570800, 3003167577200394240, 29146910264615460240
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OFFSET
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0,2
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COMMENTS
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Radius of convergence of g.f. A(x) is r = sqrt(3)/18 where A(r) = 3.
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REFERENCES
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Bruce C. Berndt, Ramanujan's Notebooks Part I, Springer Verlag, 1985, p. 305.
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LINKS
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FORMULA
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a(n) = 2^(2*n+1) * binomial(3*n/2, n) / (n+2).
A(-x) = 1/x * series reversion( x*(2*x + sqrt(1 + 4*x^2))^2 ) follows from the Lagrange inversion formula and equation 1.13, p. 305 in Berndt. Cf. A098616. - Peter Bala, Oct 19 2015
a(n) ~ 2^(n + 1/2) * 3^(3*n/2 + 1/2) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Oct 20 2015
G.f. A(x) satisfies: A(x) = 1/(1 - 4*x*A(x)^(1/2)).
G.f. A(x) satisfies: A(x) = B(x*A(x)) where B(x) = A(x/B(x)) = 1 + 8*x^2 + 4*x*sqrt(1 + 4*x^2) is the g.f. of A135863. (End)
G.f. = 1/(48*z^2) - 2F1([-2/3, -1/3], [-1/2], 108*z^2)/(48*z^2) + 4*z*2F1([5/6, 7/6],[5/2],108*z^2); a(n) = Integral_{x=0..sqrt(108)} x^n*W(x), with W(x) = ((72*(g1(x) - g2(x)) + x^2*(-g1(x) + g2(x)) + 4*sqrt(-3*x^2 + 324)*(g1(x) + g2(x)))*3^(1/6))/(96*Pi*(x^2)^(5/6)),
where g1(x) = (18 - sqrt(324 - 3*x^2))^(2/3) and
g2(x) = (18 + sqrt(324 - 3*x^2))^(2/3).
This integral representation is unique as W(x) is the solution of the Hausdorff power moment problem on x = (0, sqrt(108)). Using only the definition of a(n), W(x) can be proven to be positive. W(x) is singular at x = 0, with singularity x^(-1/3), and for x > 0 is monotonically decreasing to zero at x = sqrt(108). For x -> sqrt(108), W'(x) tends to -infinity. (End)
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EXAMPLE
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G.f.: A(x) = 1 + 4*x + 24*x^2 + 168*x^3 + 1280*x^4 + 10296*x^5 + 86016*x^6 + ... where A(x) = 1 + 4*x*A(x)^(3/2).
Radius of convergence: r = 1/(2*3^(3/2)) = 0.09622504486...
Related expansions:
A(x)^(3/2) = 1 + 6*x + 42*x^2 + 320*x^3 + 2574*x^4 + 21504*x^5 + 184756*x^6 + ...
A(x)^(1/2) = 1 + 2*x + 10*x^2 + 64*x^3 + 462*x^4 + 3584*x^5 + 29172*x^6 + ... + A078531(n)*x^n + ...
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MATHEMATICA
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Table[4^n*Binomial[3*n/2, n]*2/(n+2), {n, 0, 20}] (* Vaclav Kotesovec, Oct 20 2015 *)
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PROG
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(PARI) {a(n)=4^n*binomial(3/2*n, n)/(n/2+1)}
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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