OFFSET
0,3
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
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_{k=0..n} k*A187253(n,k).
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.
a(n) ~ 3 * 21^(3/2) * (5 + sqrt(21))^(n+1) / (Pi * n^4 * 2^(n+4)). - Vaclav Kotesovec, Dec 10 2021
EXAMPLE
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).
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(k*T(n, k), k = 0 .. n), n = 0 .. 27);
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Apr 24 2011
STATUS
approved