A246507 a(n) = 70*(n+1)*binomial(2*n+1,n+1)/(n+5).

14, 70, 300, 1225, 4900, 19404, 76440, 300300, 1178100, 4618900, 18106088, 70984095, 278369000, 1092063000, 4286142000, 16830250920, 66118842900, 259878874500, 1021939149000, 4020523757250, 15824781508536, 62313700079400, 245478212434000, 967428110493000, 3814113125277000

Offset

0,1

Comments

4*a(n+1) is the number of annular noncrossing permutations of parameter 4, see the references.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Benoît Collins, James A. Mingo, Piotr Śniady, and Roland Speicher, Second order freeness and fluctuations of random matrices. III: Higher order freeness and free cumulants, Documenta Mathematica 12 (2007), 1-70.

James A. Mingo and Alexandru Nica, Annular noncrossing permutations and partitions, and second-order asymptotics for random matrices, International Mathematics Research Notices, Vol. 2004, No. 28 (2004), pp. 1413-1460; arXiv preprint, arXiv:math/0303312 [math.OA], 2003.

Formula

O.g.f.: 2*(1-sqrt(1-4*z)-2*z-2*z^2-4*z^3-10*z^4)/(sqrt(1-4*z) *4*z^5).

Representation as the n-th moment of a signed function w(x)=2*sqrt(x)*(x^4-2*x^3-2*x^2-4*x-10)/(4*Pi*sqrt(4-x)) on the segment x=(0,4), in Maple notation: a(n) = int(x^n*w(x), x=0..4). The function w(x) -> 0 for x -> 0, and w(x) -> infinity for x->4.

a(n) ~ (35/65536)*4^n*(-755913243+151182552*n - 30236416*n^2 + 6047744*n^3 - 1212416*n^4 + 262144*n^5)/(n^(11/2)*sqrt(Pi)).

Another asymptotic series starts: a(n) ~ exp(n*log(4) + log((70*(2*n+1))/(n+5)) - log(Pi*n)/2 - 1/(8*n)). - Peter Luschny, Aug 28 2014

n*(n+5)*a(n) -2*(n+4)*(2*n+1)*a(n-1) = 0. - R. J. Mathar, Jun 14 2016

From Amiram Eldar, Feb 16 2023: (Start)

Sum_{n>=0} 1/a(n) = 2*Pi/(45*sqrt(3)) + 1/105.

Sum_{n>=0} (-1)^n/a(n) = 44*log(phi)/(175*sqrt(5)) + 1/175, where phi is the golden ratio (A001622). (End)

Mathematica

Table[70 (n+1) Binomial[2 n + 1, n + 1]/(n + 5), {n, 0, 30}] (* Vincenzo Librandi, Aug 29 2014 *)

Prog

(Magma) [70*(n+1)*Binomial(2*n+1, n+1)/(n+5): n in [0..30]]; // Vincenzo Librandi, Aug 29 2014
(PARI) for(n=0, 25, print1(70*(n+1)*binomial(2*n+1, n+1)/(n+5), ", ")) \\ G. C. Greubel, Apr 06 2017

Crossrefs

Cf. A001622, A001791, A007946, A078820.

Sequence in context: A213160 A268399 A034554 * A034562 A222989 A245950

Adjacent sequences: A246504 A246505 A246506 * A246508 A246509 A246510

Keyword

nonn

Author

Karol A. Penson, Aug 27 2014

Status

approved