A227169 a(n) = 3*((2*n+2)!)^2 / (n!*(n+1)!*(n+2)!*(n+3)!).

1, 6, 45, 392, 3780, 39204, 429429, 4907760, 58023108, 705264040, 8772399636, 111263122656, 1434941066000, 18775651948200, 248797110637125, 3333772874210400, 45115597383228900, 615974564891763000, 8477309210264363700, 117511846058893572000, 1639638607796443472400

Offset

0,2

Links

T. D. Noe, Table of n, a(n) for n = 0..200

Formula

O.g.f.: 3/(4*z^2) + (1/12288)*(-98304*z^2-2048*z+512)*EllipticK(4*sqrt(z))/(z^3*Pi) + (1/12288)*(-20480*z-512)*EllipticE(4*sqrt(z))/(z^3*Pi), in Maple notation.

E.g.f.: hypergeom([3/2, 3/2, 2], [1, 3, 4], 16*z), a 3F3 hypergeometric function.

Integral representation as the n-th moment of a signed function w(x) of bounded variation, 0 <= x <= 16: a(n) = Integral_{x=0..16}x^n*w(x) dx, n >= 0, where w(x) is the Meijer G function, w(x) = -3*MeijerG([[0], [2, 3]], [[1/2, 1/2], [1]], (1/16)*x)/Pi, satisfying w(16) = w(0) = 0, w(x) < 0 for x < 0.47.

The above Meijer G function cannot be represented by any other special function.

From R. J. Mathar, Jul 14 2013: (Start)

(n+3)*(n+2)*a(n) - 18*(n+1)^2*a(n-1) + 8*(2*n-1)^2*a(n-2) = 0.

a(n) = 3*A135389(n)/((n+2)*(n+3)) = 3*A145600(n+1)/(n+3). (End)

a(n) ~ 3 * 16^(n+1) / (Pi * n^3). - Amiram Eldar, Oct 17 2025

Maple

seq(3*((2*n+2)!)^2/(n!*(n+1)!*(n+2)!*(n+3)!), n=0..15);

Mathematica

Table[3*((2*n + 2)!)^2/(n!*(n + 1)!*(n + 2)!*(n + 3)!), {n, 0, 20}] (* T. D. Noe, Jul 12 2013 *)

Prog

(SageMath)
def a(n): return 3*(n+1)*(n+2)^2*(n+3)^3*gamma(2*n+3)^2/gamma(n+4)^4
[a(n) for n in (0..16)]  # Peter Luschny, Jul 12 2013

Crossrefs

Cf. A135389, A145600.

Sequence in context: A383616 A233668 A243694 * A365184 A135148 A137974

Adjacent sequences: A227166 A227167 A227168 * A227170 A227171 A227172

Keyword

nonn,easy

Author

Karol A. Penson, Jul 12 2013

Status

approved