A110318 Number of arcs covered by other arcs in all RNA secondary structures of size n+5 (i.e., with n+5 nodes).
1, 5, 17, 53, 157, 448, 1250, 3434, 9326, 25114, 67196, 178895, 474398, 1254072, 3306738, 8701193, 22857026, 59958380, 157098360, 411214120, 1075491286, 2810892598, 7342205478, 19168694232, 50023584613, 130497101659, 340325126923, 887307420361
Offset: 0
Keywords
Examples
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.
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
- W. R. Schmitt and M. S. Waterman, Linear trees and RNA secondary structure, Discrete Appl. Math., 51, 317-323, 1994.
- P. R. Stein and M. S. Waterman, On some new sequences generalizing the Catalan and Motzkin numbers, Discrete Math., 26 (1978), 261-272.
- M. Vauchassade de Chaumont and G. Viennot, Polynômes orthogonaux et problèmes d'énumeration en biologie moléculaire, Publ. I.R.M.A. Strasbourg, 1984, 229/S-08, Actes 8e Sem. Lotharingien, pp. 79-86.
Programs
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Maple
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);
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Mathematica
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 *)
Formula
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).
a(n) = Sum_{k>=0} k*A110317(n+5,k).