A187255 - OEIS

%I #19 Nov 11 2022 08:07:06

%S 0,1,2,4,10,29,88,281,928,3170,11108,39844,145700,542099,2047390,

%T 7837154,30359390,118877686,470023034,1874858180,7538907394,

%U 30538466279,124544812652,511112081854,2109676656860,8754763717529,36512355688438,152987775947396

%N Number of isolated vertices in all 3-noncrossing RNA structures on n vertices.

%H Vaclav Kotesovec, <a href="/A187255/b187255.txt">Table of n, a(n) for n = 0..1000</a>

%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_{k=0..n} k*A187253(n,k).

%F a(n) = Sum_{k=0..n} k*T(n,k), where T(n,k) = Sum_{j=0..(n-k)/2} (-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), and c(n)=A000108(n) are the Catalan numbers.

%F a(n) ~ 3 * 21^(3/2) * (5 + sqrt(21))^(n+1) / (Pi * n^4 * 2^(n+4)). - _Vaclav Kotesovec_, Dec 10 2021

%e a(4)=10 because in ABAB, AIAI, AIIA, IAIA, and IIII, where pairs of A's and pairs of B's are assumed to be joined by an arc, we have a total of 0+2+2+2+4=10 isolated vertices (the I's).

%p 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(k*T(n, k), k = 0 .. n), n = 0 .. 27);

%Y Cf. A187253, A000108.

%K nonn

%O 0,3

%A _Emeric Deutsch_, Apr 24 2011