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%I #39 Nov 15 2022 16:09:14
%S 1,0,1,4,22,139,979,7484,61018,523995,4696277,43623618,417729564,
%T 4106089683,41289287337,423556384020,4422308778458,46904447607369,
%U 504544306691569,5496706186024364,60576765646658782,674624324569952719,7585425185883023881
%N Number of 3-noncrossing RNA structures over 2n vertices with no isolated vertices.
%H Emma Y. Jin, Jing Qin and Christian M. Reidys, <a href="https://arxiv.org/abs/0704.2518">Combinatorics of RNA structures with pseudoknots</a>, arXiv:0704.2518 [math.CO], 2007.
%H Emma Y. Jin, Jing Qin and Christian M. Reidys, <a href="https://doi.org/10.1007/s11538-007-9240-y">Combinatorics of RNA structures with pseudoknots</a>, Bulletin of Mathematical Biology Vol. 70 (2008) pp. 45-67.
%F 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.
%F a(n) = A187253(2*n, 0).
%F a(n) ~ 27 * (1 + sqrt(3))^(4*n + 2) / (Pi * n^5 * 2^(2*n + 4)). - _Vaclav Kotesovec_, Dec 10 2021
%F 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
%F 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
%F a(n) = A358119(n) - A358118(n), (see first formula). - _Peter Luschny_, Nov 11 2022
%F 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
%e a(3) = 4 because we have ABACBC, ABCBAC, ABCACB, and ABCBCA, where identically labeled vertices are assumed to be joined by an arc.
%p 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);
%t 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 *)
%Y Cf. A000108, A187253, A358118, A358119.
%K nonn
%O 0,4
%A _Emeric Deutsch_, Apr 24 2011
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