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A110317 Triangle read by rows: T(n,k) (k>=0) is the number of RNA secondary structures of size n (i.e., with n nodes) having k arcs that are covered by other arcs. 1
1, 1, 1, 2, 4, 7, 1, 12, 5, 21, 15, 1, 37, 37, 8, 65, 84, 35, 1, 114, 182, 115, 12, 200, 381, 323, 73, 1, 351, 777, 825, 313, 17, 616, 1554, 1977, 1087, 138, 1, 1081, 3062, 4524, 3291, 754, 23, 1897, 5962, 9999, 9063, 3209, 241, 1, 3329, 11496, 21515, 23300 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Rows 0,1,2 have one term each; row n >= 3 has ceiling(n/2) - 1 terms.

Rows sums yield A004148.

T(n,0) = A005251(n+1).

Sum_{k>=0} k*T(n,k) = A110318(n-5).

LINKS

Table of n, a(n) for n=0..55.

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.

FORMULA

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

EXAMPLE

T(6,1)=5 because we have 15/(24)/3/6, 16/(24)/3/5, 16/(25)/3/4, 16/2/(35)/4 and 1/26/(35)/4 (the covered arcs are shown between parentheses).

Triangle begins

   1;

   1;

   1;

   2;

   4;

   7,  1;

  12,  5;

  21, 15,  1;

  37, 37,  8;

MAPLE

G:=2*t/(2*t-2*z*t-1+z+t*z^2+sqrt(1-2*z-2*t*z^2+z^2-2*t*z^3+t^2*z^4)): Gser:=simplify(series(G, z=0, 20)): P[0]:=1: for n from 1 to 17 do P[n]:=coeff(Gser, z^n) od: for n from 0 to 2 do print(1) od: for n from 3 to 17 do seq(coeff(t*P[n], t^k), k=1..ceil(n/2)-1) od; # yields sequence in triangular form

CROSSREFS

Cf. A004148, A005252, A110318.

Sequence in context: A089349 A118424 A118429 * A098073 A118390 A247294

Adjacent sequences:  A110314 A110315 A110316 * A110318 A110319 A110320

KEYWORD

nonn,tabf

AUTHOR

Emeric Deutsch, Jul 19 2005

STATUS

approved

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Last modified August 21 08:34 EDT 2019. Contains 326162 sequences. (Running on oeis4.)