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A133365
Number of 3-noncrossing RNA structures, i.e., the number of 3-noncrossing partial matchings over n vertices and without arcs of length 1.
2
1, 1, 2, 5, 13, 36, 105, 321, 1018, 3334, 11216, 38635, 135835, 486337, 1769500, 6531796, 24425758, 92420026, 353444218, 1364933719, 5318450239, 20894505025, 82713826842, 329746065427, 1323179962753, 5341963415921, 21689519880470, 88533441655211
OFFSET
1,3
COMMENTS
a(n) is the sum of entries in row n of the triangle in A187253.
a(n) is asymptotically equal to 4!*10.4724*((5+sqrt(21))/2)^n/(n(n-1)(n-2)(n-3)(n-4)).
LINKS
Emma Y. Jin, Jing Qin and Christian M. Reidys, Combinatorics of RNA structures with pseudoknots, arXiv:0704.2518 [math.CO], 2007.
Emma Y. Jin, Jing Qin and Christian M. Reidys, Combinatorics of RNA structures with pseudoknots, Bulletin of Mathematical Biology Vol. 70 (2008) pp. 45-67.
Emma Y. Jin and Christian M. Reidys, Asymptotic Enumeration of RNA Structures with Pseudoknots, arXiv:0706.3137 [q-bio.BM], 2007.
Emma Y. Jin and Christian M. Reidys, Asymptotic Enumeration of RNA Structures with Pseudoknots, Bulletin of Mathematical Biology 70 (2008), 951-970.
Emma Y. Jin and Christian M. Reidys, Central and local limit theorems for RNA structures, J. Theoretical Biology, 250, 2008, 547-559.
FORMULA
a(n) = Sum_{k=0..n} T(n,k), where T(n,k) = Sum((-1)^j*binomial(n-j,j)*binomial(n-2j,k)*[c((n-k)/2-2j)*c((n-k)/2-j+2)-c((n-k)/2-j+1)^2], j=0..(n-k)/2), and c(n)=A000108(n) are the Catalan numbers. [Perhaps this formula is using the convention that c(x) = 0 unless x is a nonnegative integer? - N. J. A. Sloane, Jul 24 2017]
EXAMPLE
a(4)=5 because we have ABAB, AIAI, AIIA, IAIA, and IIII, where pairs of A's and pairs of B's are assumed to be joined by an arc and the I's are isolated vertices.
MAPLE
c := proc (n) options operator, arrow: binomial(2*n, n)/(n+1) end proc: T := proc (n, k) if `mod`(n-k, 2) = 0 then sum((-1)^b*binomial(n-b, b)*binomial(n-2*b, k)*(c((1/2)*n-(1/2)*k-b)*c((1/2)*n-(1/2)*k-b+2)-c((1/2)*n-(1/2)*k-b+1)^2), b = 0 .. (1/2)*n-(1/2)*k) else 0 end if end proc: seq(add(T(n, k), k = 0 .. n), n = 1 .. 28);
MATHEMATICA
c = CatalanNumber;
T[n_, k_] := If[EvenQ[m = n-k], Sum[(-1)^b*Binomial[n-b, b] * Binomial[n - 2*b, k] * (c[m/2-b]*c[m/2-b+2] - c[m/2-b+1]^2), {b, 0, m/2}], 0];
a[n_] := Sum[T[n, k], {k, 0, n}];
Array[a, 28] (* Jean-François Alcover, Nov 26 2017, from Maple *)
CROSSREFS
Sequence in context: A114465 A135310 A135337 * A370886 A135335 A336989
KEYWORD
nonn
AUTHOR
Emma Y. Jin (emma(AT)cfc.nankai.edu.cn), Oct 26 2007
STATUS
approved