|
%I #16 Jun 13 2017 06:51:21
%S 1,5,17,53,157,448,1250,3434,9326,25114,67196,178895,474398,1254072,
%T 3306738,8701193,22857026,59958380,157098360,411214120,1075491286,
%U 2810892598,7342205478,19168694232,50023584613,130497101659,340325126923,887307420361
%N Number of arcs covered by other arcs in all RNA secondary structures of size n+5 (i.e., with n+5 nodes).
%H Vincenzo Librandi, <a href="/A110318/b110318.txt">Table of n, a(n) for n = 0..1000</a>
%H W. R. Schmitt and M. S. Waterman, <a href="http://dx.doi.org/10.1016/0166-218X(92)00038-N">Linear trees and RNA secondary structure</a>, Discrete Appl. Math., 51, 317-323, 1994.
%H P. R. Stein and M. S. Waterman, <a href="http://dx.doi.org/10.1016/0012-365X(79)90033-5">On some new sequences generalizing the Catalan and Motzkin numbers</a>, Discrete Math., 26 (1978), 261-272.
%H M. Vauchassade de Chaumont and G. Viennot, <a href="http://www.emis.de/journals/SLC/opapers/s08viennot.pdf">Polynômes orthogonaux et problèmes d'énumeration en biologie moléculaire</a>, Publ. I.R.M.A. Strasbourg, 1984, 229/S-08, Actes 8e Sem. Lotharingien, pp. 79-86.
%F G.f.: 2(1-2z-z^3-(1-z)Q)/(z^5*Q(1-z+z^2+Q)^2), where Q:=sqrt(1-2z-z^2-2z^3+z^4).
%F a(n) = Sum_{k>=0} k*A110317(n+5,k).
%e a(0)=1 because in the 8 (=A004148(5)) RNA secondary structures of size 5, namely 1/2/3/4/5, 13/2/4/5, 14/2/3/5, 15/2/3/4, 1/24/3/5, 1/25/3/4, 1/2/35/4 and 15/24/3 we have altogether 1 arc covered by another arc: in 15/24/3 the arc 24 is covered by the arc 15.
%p Q:=sqrt(1-2*z-z^2-2*z^3+z^4): G:=2*(1-2*z-z^3-(1-z)*Q)/Q/z^5/(1-z+z^2+Q)^2: Gser:=series(G,z=0,38): 1,seq(coeff(Gser,z^n),n=1..30);
%t CoefficientList[Series[2 (1 - 2 x - x^3 - (1 - x) Sqrt[1 - 2 x - x^2 - 2 x^3 + x^4]) / (x^5 Sqrt[1 - 2 x - x^2 - 2 x^3 + x^4] (1 - x + x^2 + Sqrt[1 - 2 x - x^2 - 2 x^3 + x^4])^2), {x, 0, 33}], x] (* _Vincenzo Librandi_, Jun 13 2017 *)
%Y Cf. A004148, A110317.
%K nonn
%O 0,2
%A _Emeric Deutsch_, Jul 19 2005
|
