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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 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-François 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.)