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A088518 Symmetric secondary structures of RNA molecules with n nucleotides. 4
1, 1, 1, 2, 2, 4, 5, 9, 12, 21, 29, 50, 71, 121, 175, 296, 434, 730, 1082, 1812, 2709, 4521, 6807, 11328, 17157, 28485, 43359, 71844, 109830, 181674, 278769, 460443, 708840, 1169283, 1805291, 2974574, 4604363, 7578937, 11758552, 19337489, 30064037 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Diagonal sums of triangle in A088855. [From Philippe Deléham, Jan 04 2009]

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

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

FORMULA

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.

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

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

In closed form, c = sqrt(1+3/sqrt(5)) / sqrt(Pi) = 0.863346635039540133... - Vaclav Kotesovec, Mar 21 2014

MAPLE

b:= proc(n) option remember;

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

    end:

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

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

    end:

seq(a(n), n=0..50);  # Alois P. Heinz, Aug 27 2012

MATHEMATICA

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 *)

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 *)

CROSSREFS

Cf. A004148.

Sequence in context: A038000 A204856 A124280 * A001224 A102526 A050192

Adjacent sequences:  A088515 A088516 A088517 * A088519 A088520 A088521

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Nov 18 2003

STATUS

approved

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Last modified May 26 18:43 EDT 2017. Contains 287129 sequences.