login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A110320 Number of blocks in all RNA secondary structures with n nodes (an RNA secondary structure can be viewed as a restricted noncrossing partition). 17
1, 2, 5, 13, 32, 80, 201, 505, 1273, 3217, 8146, 20668, 52531, 133726, 340909, 870213, 2223958, 5689807, 14571335, 37350585, 95821071, 246015677, 632088930, 1625119218, 4180845277, 10762096850, 27718352411, 71426753423, 184146711578 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Antidiagonal sums of A132812. - Philippe Deléham, Jun 08 2013

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..1000

Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.

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, Polynomes 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.

FORMULA

G.f.: (1-z-z^2)/(2*z^2*sqrt(1-2*z-z^2-2*z^3+z^4))-1/(2*z^2).

a(n) = Sum_{k=1..n} k*A110319(n,k).

Conjecture: a(n) = (A051292(n+2)-A051286(n+1))/2. - Gerald McGarvey, Jan 14 2007

Conjecture: (2*n-1)*(2*n-3)*(n+2)*a(n) -4*n*(n+1)*(2*n-3)*a(n-1) -n*(4*n^2-4*n+5)*a(n-2) -2*(2*n+1)*(2*n^2-4*n+1)*a(n-3) +(n-2)*(2*n+1)*(2*n-1)*a(n-4)=0. - R. J. Mathar, Sep 02 2014

a(n) = (A051286(n+2)-A051286(n+1)-A051286(n))/2. - Benedict W. J. Irwin, Sep 24 2016

a(n) ~ sqrt(4 + 9/sqrt(5)) * (3+sqrt(5))^n / (sqrt(Pi*n) * 2^(n+1)). - Vaclav Kotesovec, Sep 25 2016

EXAMPLE

a(4)=13 because the 4 (=A004148(4)) RNA secondary structures of size 4, namely 1/2/3/4, 13/2/4, 14/2/3 and 1/24/3, have altogether 4+3+3+3=13 blocks.

MAPLE

G:=1/2*(1-z-z^2)/z^2/(1-2*z-z^2-2*z^3+z^4)^(1/2)-1/2*1/(z^2): Gser:=series(G, z=0, 37): seq(coeff(Gser, z^n), n=1..33);

MATHEMATICA

Table[Sum[Binomial[n-j+1, j]Binomial[n-j+1, j-1], {j, 0, n}], {n, 1, 25}] (* Benedict W. J. Irwin, Sep 24 2016 *)

CROSSREFS

Cf. A004148, A110319.

Sequence in context: A255170 A255630 A298535 * A219230 A108890 A220739

Adjacent sequences:  A110317 A110318 A110319 * A110321 A110322 A110323

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Jul 19 2005

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 15 21:06 EDT 2018. Contains 316237 sequences. (Running on oeis4.)