

A187253


Triangle read by rows: T(n,k) is the number of 3noncrossing RNA structures on n vertices having k isolated vertices.


3



1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 3, 0, 1, 0, 6, 0, 6, 0, 1, 4, 0, 21, 0, 10, 0, 1, 0, 34, 0, 55, 0, 15, 0, 1, 22, 0, 157, 0, 120, 0, 21, 0, 1, 0, 232, 0, 526, 0, 231, 0, 28, 0, 1, 139, 0, 1317, 0, 1435, 0, 406, 0, 36, 0, 1, 0, 1761, 0, 5355, 0, 3388, 0, 666, 0, 45, 0, 1, 979, 0, 11883, 0, 17500, 0, 7182, 0, 1035, 0, 55, 0, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,13


COMMENTS

Sum of entries in row n is A133365(n).
T(n,k)=0 if nk is odd.
T(n,0)=A187254(n).
Sum(k*T(n,k), k=0..n)=A187255(n).


REFERENCES

Emma Y. Jin, Jing Qin and Christian M. Reidys, "Combinatorics of RNA Structures with Pseudoknots", Bulletin of Mathematical Biology Vol. 70 (2008) pp. 4567.


LINKS

Table of n, a(n) for n=0..90.


FORMULA

T(n,k)=Sum((1)^j*binom(nj,j)*binom(n2j,k)*[c((nk)/22j)*c((nk)/2j+2)c((nk)/2j+1)^2], j=0..(nk)/2), where c(n)=A000108(n) are the Catalan numbers (see Corollary 2 in the Jin et al. reference.


EXAMPLE

T(4,2)=3 because we have AIAI, IAIA, AIIA, where in each structure the two A's are joined by an arc and the two I's are isolated vertices.
T(4,4)=1 because we have IIII.
T(4,0)=1 because we have ABAB, where the two A's are joined by an arc and the two B's are joined by an arc.
Triangle starts:
1;
0,1;
0,0,1;
0,1,0,1;
1,0,3,0,1;
0,6,0,6,0,1;
4,0,21,0,10,0,1.


MAPLE

c := proc (n) options operator, arrow: binomial(2*n, n)/(n+1) end proc: T := proc (n, l) if `mod`(nl, 2) = 0 then sum((1)^b*binomial(nb, b)*binomial(n2*b, l)*(c((1/2)*n(1/2)*lb)*c((1/2)*n(1/2)*lb+2)c((1/2)*n(1/2)*lb+1)^2), b = 0 .. (1/2)*n(1/2)*l) else 0 end if end proc: for n from 0 to 12 do seq(T(n, k), k = 0 .. n) end do; # yields sequence in triangular form


CROSSREFS

Cf. A000108, A133365, A187254, A187255.
Sequence in context: A136689 A073278 A081658 * A022904 A238341 A242451
Adjacent sequences: A187250 A187251 A187252 * A187254 A187255 A187256


KEYWORD

nonn,tabl


AUTHOR

Emeric Deutsch, Apr 24 2011


STATUS

approved



