login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A214377 G.f. satisfies: A(x) = 1 + 4*x*A(x)^(3/2). 6
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

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

License Agreements, Terms of Use, Privacy Policy .

Last modified February 20 21:53 EST 2018. Contains 299387 sequences. (Running on oeis4.)