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URL: https://oeis.org/A187254

⇱ A187254 - OEIS

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A187254
Number of 3-noncrossing RNA structures over 2n vertices with no isolated vertices.
1
1, 0, 1, 4, 22, 139, 979, 7484, 61018, 523995, 4696277, 43623618, 417729564, 4106089683, 41289287337, 423556384020, 4422308778458, 46904447607369, 504544306691569, 5496706186024364, 60576765646658782, 674624324569952719, 7585425185883023881
OFFSET
0,4
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.
FORMULA
a(n) = Sum_{j=0..n} (-1)^j*binomial(2n-j,j)*(c(n-j)*c(n-j+2) - c(n-j+1)^2), where c(i) = A000108(i) are the Catalan numbers.
a(n) = A187253(2*n, 0).
a(n) ~ 27 * (1 + sqrt(3))^(4*n + 2) / (Pi * n^5 * 2^(2*n + 4)). - Vaclav Kotesovec, Dec 10 2021
D-finite with recurrence (n+3)*(n+2)*a(n) + 2*(-7*n^2-2)*a(n-1) + 2*(-2*n+3)*a(n-2) + 2*(7*n^2-42*n+65)*a(n-3) - (n-5)*(n-6)*a(n-4) = 0. - R. J. Mathar, Jul 22 2022
G.f.: (1/4)*(x + 9 - (1 - 14*x + x^2)^(3/2)/x^2*hypergeom([-3/2, 5/2], [2], -16*x/(1 - 14*x + x^2))). - Mark van Hoeij, Nov 10 2022
a(n) = A358119(n) - A358118(n), (see first formula). - Peter Luschny, Nov 11 2022
Maple (depending on the version) gives the third-order recurrence (n - 5)*(2*n - 1)*(n - 4)*a(n - 3) - (n - 1)*(13*n - 24)*(2*n - 3)*a(n - 2) - (n - 1)*(2*n - 1)*(13*n - 2)*a(n - 1) + (n + 3)*(n + 2)*(2*n - 3)*a(n) = 0. - Peter Bala, Nov 11 2022
EXAMPLE
a(3) = 4 because we have ABACBC, ABCBAC, ABCACB, and ABCBCA, where identically labeled vertices are assumed to be joined by an arc.
MAPLE
c := n -> binomial(2*n, n)/(n + 1): a := n -> add((-1)^j*binomial(2*n-j, j)*(c(n-j)*c(n-j+2) - c(n-j+1)^2), j = 0..n): seq(a(n), n = 0 .. 22);
MATHEMATICA
Table[Sum[(-1)^j*Binomial[2*n-j, j] * (CatalanNumber[n-j]*CatalanNumber[n-j+2] - CatalanNumber[n-j+1]^2), {j, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Dec 10 2021 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Apr 24 2011
STATUS
approved