A289031 Number of perfect matchings on n+3 edges which represent RNA secondary folding structures characterized by the Reeder and Giegerich and the Lyngso and Pedersen families, but not the family characterized by Cao and Chen.

1, 8, 42, 186, 759, 2970, 11369, 43024, 161889, 607630, 2279156, 8552292, 32124073, 120828404, 455175495, 1717506346, 6491412107, 24575174688, 93187097419, 353912403794, 1346146363275, 5127660231072, 19559151930621, 74706450932970

Offset: 1

Author

Kyle Goryl, Jun 22 2017

Keywords

• nonn

Comments

These matchings can be created inductively by beginning with a hairpin that has a single edge inserted into its middle, then inserting noncrossing matchings into the matching. Finally we can inflate the edges of the hairpin by ladders.

Links

• Aziza Jefferson The Substitution Decomposition of Matchings and RNA Secondary Structures, PhD Thesis, University of Florida, 2015.

Crossrefs

Cf. A003517.

Formula

a(n) = 2*a(n-1) - a(n-2) + A003517(n+1).

D-finite recurrence: (n-1)*(n+5)*a(n) = 2*(3*n^2 + 9*n - 2)*a(n-1) - (3*n + 1)*(3*n + 7)*a(n-2) + 2*(n+1)*(2*n + 3)*a(n-3). - Vaclav Kotesovec, Jun 24 2017

a(n) ~ 2^(2*n+8) / (3*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jun 27 2017