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A103140
Number of 3-noncrossing restricted RNA structures with n vertices.
0
1, 1, 1, 2, 5, 14, 40, 119, 364, 1145, 3688, 12139, 40734, 139071, 482214, 1695469, 6036768, 21740969, 79117822, 290674470, 1077306351, 4025068621, 15151115808, 57427176992, 219068962330, 840708048210, 3244438898552, 12586627632549, 49069788882951
OFFSET
1,4
COMMENTS
a(n) is the number of 3-noncrossing partial matchings over n vertices and without arcs of length 1 and 2. - Andrey Zabolotskiy, Nov 11 2023
LINKS
Andrei Asinowski, Axel Bacher, Cyril Banderier, and Bernhard Gittenberger, Analytic combinatorics of lattice paths with forbidden patterns, the vectorial kernel method, and generating functions for pushdown automata, Laboratoire d'Informatique de Paris Nord (LIPN 2019).
Emma Y. Jin, Jing Qin and Christian M. Reidys, Combinatorics of RNA structures with pseudoknots, arXiv:0704.2518 [math.CO], 2007.
Emma Y. Jin, Jing Qin and Christian M. Reidys, Combinatorics of RNA structures with pseudoknots, Bulletin of Mathematical Biology Vol. 70 (2008) pp. 45-67. See Table 2 on page 62 for details.
MATHEMATICA
sf3[n_] := sf3[n] = Sum[Binomial[n, 2 k] (CatalanNumber[k + 2] CatalanNumber[k] - CatalanNumber[k + 1]^2), {k, 0, n/2}]; (* this is A049401 *)
la[0, 0, 0] = 1;
la[_?Negative, _, _] = la[_, _?Negative, _] = la[_, _, _?Negative] = 0;
la[n_, b1_, b2_] := la[n, b1, b2] = la[n - 2, b1 - 1, b2] + la[n - 1, b1, b2] + la[n - 4, b1, b2 - 2] + la[n - 3, b1, b2 - 1];
a[n_] := Sum[(-1)^(b1 + b2) la[n, b1, b2] sf3[n - 2 (b1 + b2)], {b1, 0, n/2}, {b2, 0, n/2}];
Table[a[n], {n, 30}] (* Andrey Zabolotskiy, Nov 11 2023, from eqs. (4.2), (4.3), and (2.14) by Jin et al. *)
CROSSREFS
Sequence in context: A349413 A200438 A363933 * A148320 A374541 A301871
KEYWORD
nonn
AUTHOR
Parthasarathy Nambi, Sep 07 2008
EXTENSIONS
Terms a(16) and beyond from Andrey Zabolotskiy, Nov 11 2023
STATUS
approved