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%I #39 Feb 16 2023 05:08:04
%S 14,70,300,1225,4900,19404,76440,300300,1178100,4618900,18106088,
%T 70984095,278369000,1092063000,4286142000,16830250920,66118842900,
%U 259878874500,1021939149000,4020523757250,15824781508536,62313700079400,245478212434000,967428110493000,3814113125277000
%N a(n) = 70*(n+1)*binomial(2*n+1,n+1)/(n+5).
%C 4*a(n+1) is the number of annular noncrossing permutations of parameter 4, see the references.
%H Vincenzo Librandi, <a href="/A246507/b246507.txt">Table of n, a(n) for n = 0..1000</a>
%H Benoît Collins, James A. Mingo, Piotr Śniady, and Roland Speicher, <a href="https://doi.org/10.4171/dm/220">Second order freeness and fluctuations of random matrices. III: Higher order freeness and free cumulants</a>, Documenta Mathematica 12 (2007), 1-70.
%H James A. Mingo and Alexandru Nica, <a href="https://doi.org/10.1155/S1073792804133023">Annular noncrossing permutations and partitions, and second-order asymptotics for random matrices</a>, International Mathematics Research Notices, Vol. 2004, No. 28 (2004), pp. 1413-1460; <a href="http://arxiv.org/abs/math/0303312">arXiv preprint</a>, arXiv:math/0303312 [math.OA], 2003.
%F 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).
%F 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.
%F 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)).
%F 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
%F n*(n+5)*a(n) -2*(n+4)*(2*n+1)*a(n-1) = 0. - _R. J. Mathar_, Jun 14 2016
%F From _Amiram Eldar_, Feb 16 2023: (Start)
%F Sum_{n>=0} 1/a(n) = 2*Pi/(45*sqrt(3)) + 1/105.
%F Sum_{n>=0} (-1)^n/a(n) = 44*log(phi)/(175*sqrt(5)) + 1/175, where phi is the golden ratio (A001622). (End)
%t Table[70 (n+1) Binomial[2 n + 1, n + 1]/(n + 5), {n, 0, 30}] (* _Vincenzo Librandi_, Aug 29 2014 *)
%o (Magma) [70*(n+1)*Binomial(2*n+1,n+1)/(n+5): n in [0..30]]; // _Vincenzo Librandi_, Aug 29 2014
%o (PARI) for(n=0,25, print1(70*(n+1)*binomial(2*n+1,n+1)/(n+5), ", ")) \\ _G. C. Greubel_, Apr 06 2017
%Y Cf. A001622, A001791, A007946, A078820.
%K nonn
%O 0,1
%A _Karol A. Penson_, Aug 27 2014
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