A187255 Number of isolated vertices in all 3-noncrossing RNA structures on n vertices.

0, 1, 2, 4, 10, 29, 88, 281, 928, 3170, 11108, 39844, 145700, 542099, 2047390, 7837154, 30359390, 118877686, 470023034, 1874858180, 7538907394, 30538466279, 124544812652, 511112081854, 2109676656860, 8754763717529, 36512355688438, 152987775947396

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

Cf. A187253, A000108.

Sequence in context: A243814 A005505 A262592 * A148114 A135334 A182486

Adjacent sequences: A187252 A187253 A187254 * A187256 A187257 A187258

Keyword

nonn

Author

Emeric Deutsch, Apr 24 2011

Status

approved