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A103140
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Number of 3-noncrossing restricted RNA structures with n vertices.
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0
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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
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OFFSET
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1,4
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COMMENTS
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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
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LINKS
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MATHEMATICA
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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. *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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