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 A214377 G.f. satisfies: A(x) = 1 + 4*x*A(x)^(3/2). 7
 1, 4, 24, 168, 1280, 10296, 86016, 739024, 6488064, 57946200, 524812288, 4808643120, 44493176832, 415146189360, 3901709352960, 36902658748320, 350980432461824, 3354743017001880, 32207616155320320, 310446853795570800, 3003167577200394240, 29146910264615460240 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Radius of convergence of g.f. A(x) is r = sqrt(3)/18 where A(r) = 3. REFERENCES Bruce C. Berndt, Ramanujan's Notebooks Part I, Springer Verlag, 1985, p. 305. LINKS G. C. Greubel, Table of n, a(n) for n = 0..975 FORMULA a(n) = 2^(2*n+1) * binomial(3*n/2, n) / (n+2). Self-convolution of A078531. 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 EXAMPLE 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 +... MATHEMATICA Table[4^n*Binomial[3*n/2, n]*2/(n+2), {n, 0, 20}] (* Vaclav Kotesovec, Oct 20 2015 *) PROG (PARI) {a(n)=4^n*binomial(3/2*n, n)/(n/2+1)} for(n=0, 30, print1(a(n), ", ")) CROSSREFS Cf. A078531, A214553, A098616. Sequence in context: A213441 A238299 A221656 * A212277 A246423 A188913 Adjacent sequences:  A214374 A214375 A214376 * A214378 A214379 A214380 KEYWORD nonn AUTHOR Paul D. Hanna, Jul 14 2012 STATUS approved

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Last modified October 20 07:17 EDT 2018. Contains 316378 sequences. (Running on oeis4.)