The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A214377 G.f. satisfies: A(x) = 1 + 4*x*A(x)^(3/2). 8
 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 Alois Panholzer, Parking function varieties for combinatorial tree models, arXiv:2007.14676 [math.CO], 2020. 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 G.f.: 4*x*(sin(asin(216*x^2-1)/3)/(6*x)+1/(12*x))^3+1. - Vladimir Kruchinin, Sep 30 2022 From Paul D. Hanna, Feb 03 2023: (Start) 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) From Karol A. Penson, Mar 23 2024: (Start) 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) 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, A135863, A214553, A098616. Sequence in context: A221656 A366623 A339346 * A331007 A369503 A212277 Adjacent sequences: A214374 A214375 A214376 * A214378 A214379 A214380 KEYWORD nonn AUTHOR Paul D. Hanna, Jul 14 2012 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified September 12 18:50 EDT 2024. Contains 375853 sequences. (Running on oeis4.)