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A088518 Symmetric secondary structures of RNA molecules with n nucleotides. 4

%I

%S 1,1,1,2,2,4,5,9,12,21,29,50,71,121,175,296,434,730,1082,1812,2709,

%T 4521,6807,11328,17157,28485,43359,71844,109830,181674,278769,460443,

%U 708840,1169283,1805291,2974574,4604363,7578937,11758552,19337489,30064037

%N Symmetric secondary structures of RNA molecules with n nucleotides.

%C Diagonal sums of triangle in A088855. [From _Philippe Deléham_, Jan 04 2009]

%C Number of prime symmetric Dyck (n+2)-paths with no ascent of length 1. E.g. the a(3) = 2 5-paths are UUUUUDDDDD and UUUDDUUDDD. - _David Scambler_, Aug 27 2012

%H Alois P. Heinz, <a href="/A088518/b088518.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f. H(z) satisfies z^2*(1-z-z^2)H^2+(1-z-z^2)(1+z-z^2)H-(1+z-z^2)=0. H=[1/(1-z-z^2)]C(-z^2/(1-3z^2+z^4)), where C(z)=(1-sqrt(1-4z))/(2z) is the Catalan function. a(0)=a(1)=1; a(2n)=a(2n-1)+a(2n-2)-A004148(n-1) for n > 0; a(2n+1)=a(2n)+a(2n-1) for n > 0.

%F a(n) = F(n) - Sum[A004148(i)*F(n-1-2i), i=1..floor(n/2)-1], where F(i)=A000045(i) are the Fibonacci numbers. - _Emeric Deutsch_, Nov 19 2003

%F a(n) is asymptotic to c*phi^n/sqrt(n) where phi=(1+sqrt(5))/2 and c=0.86.... - _Benoit Cloitre_, Nov 19 2003

%F In closed form, c = sqrt(1+3/sqrt(5)) / sqrt(Pi) = 0.863346635039540133... - _Vaclav Kotesovec_, Mar 21 2014

%p b:= proc(n) option remember;

%p `if`(n=0, 1, b(n-1)+ add(b(k)*b(n-2-k), k=1..n-2))

%p end:

%p a:= proc(n) option remember; `if`(n<2, 1,

%p a(n-1) +a(n-2) +`if`(irem(n, 2, 'r')=0, -b(r-1), 0))

%p end:

%p seq(a(n), n=0..50); # _Alois P. Heinz_, Aug 27 2012

%t CoefficientList[Series[(1 - 3*x^2 + x^4 - Sqrt[1 - 2*x^2 - x^4 - 2*x^6 + x^8])/(2*x^2*(-1 + x + x^2)), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Mar 21 2014 *)

%t b[n_] := b[n] = If[n==0, 1, b[n-1] + Sum[b[k]*b[n-2-k], {k, 1, n-2}]]; a[n_] := a[n] = If[n<2, 1, a[n-1] + a[n-2] + If[{q, r} = QuotientRemainder[n, 2 ]; r==0, -b[q-1], 0]]; Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Mar 31 2015, after _Alois P. Heinz_ *)

%Y Cf. A004148.

%K nonn

%O 0,4

%A _Emeric Deutsch_, Nov 18 2003

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Last modified September 15 12:38 EDT 2019. Contains 327078 sequences. (Running on oeis4.)