A289834 Number of perfect matchings on n edges which represent RNA secondary folding structures characterized by the Lyngso and Pedersen (L&P) family and the Cao and Chen (C&C) family.

1, 1, 3, 11, 39, 134, 456, 1557, 5364, 18674, 65680, 233182, 834796, 3010712, 10929245, 39904623, 146451871, 539972534, 1999185777, 7429623640, 27705320423, 103636336176, 388775988319, 1462261313876, 5513152229901, 20832701135628, 78884459229627

Offset: 0

Author

Kyle Goryl, Jul 13 2017

Keywords

• nonn

Links

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

Crossrefs

Cf. A000108, A256334.

Programs

Maple

a:= proc(n) option remember; `if`(n<4, [1$2, 3, 11][n+1],
      (2*(74*n^2-69*n-110)*a(n-1)-3*(89*n^2-139*n-70)*a(n-2)+
       2*(91*n^2-204*n-52)*a(n-3)-4*(5*n+1)*(2*n-7)*a(n-4))
       /((n+2)*(23*n-43)))
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Jul 13 2017

Mathematica

c[n_] := c[n] = CatalanNumber[n];
b[n_] := b[n] = If[n<2, 0, 2+((5n-9) b[n-1] - (4n-2) b[n-2])/(n-1)];
a[n_] := Sum[c[i] Sum[c[j]-(n-i), {j, 1, n-i}], {i, 0, n-2}] + b[n] + c[n];
a /@ Range[0, 40] (* Jean-François Alcover, Nov 29 2020 *)

Python

from functools import cache
@cache
def a(n):
    return (
        [1, 1, 3, 11][n]
        if n < 4
        else (
              2 * (74 * n ** 2 - 69 * n - 110) * a(n - 1)
            - 3 * (89 * n ** 2 - 139 * n - 70) * a(n - 2)
            + 2 * (91 * n ** 2 - 204 * n - 52) * a(n - 3)
            - 4 * (5 * n + 1) * (2 * n - 7) * a(n - 4)
        )
        // ((n + 2) * (23 * n - 43))
    )
print([a(n) for n in range(27)])
# Indranil Ghosh, Jul 15 2017, after Maple code, updated by Peter Luschny, Nov 29 2020

Formula

a(n) = Sum_{i=0..n-2} C_i*(Sum_{j=1..n-i} C_j - (n-i)) + C_n where C is A000108.

From Vaclav Kotesovec, Jul 13 2017: (Start)

D-finite recurrence (of order 3): (n+2)*(41*n^3 - 228*n^2 + 391*n - 180)*a(n) = 6*(41*n^4 - 187*n^3 + 192*n^2 + 120*n - 160)*a(n-1) - 3*(3*n - 4)*(41*n^3 - 146*n^2 + 83*n + 70)*a(n-2) + 2*(2*n - 5)*(41*n^3 - 105*n^2 + 58*n + 24)*a(n-3).

a(n) ~ 41 * 4^n / (9*sqrt(Pi)*n^(3/2)).

(End)

Extensions

More terms from Alois P. Heinz, Jul 13 2017