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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. 0
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Table of n, a(n) for n=0..26.

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

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)

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-Fran├žois Alcover, Nov 29 2020 *)

PROG

(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

CROSSREFS

Cf. A000108, A256334.

Sequence in context: A227638 A166336 A002783 * A007482 A134760 A257290

Adjacent sequences:  A289831 A289832 A289833 * A289835 A289836 A289837

KEYWORD

nonn

AUTHOR

Kyle Goryl, Jul 13 2017

EXTENSIONS

More terms from Alois P. Heinz, Jul 13 2017

STATUS

approved

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Last modified August 13 16:07 EDT 2022. Contains 356107 sequences. (Running on oeis4.)