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A187253 Triangle read by rows: T(n,k) is the number of 3-noncrossing 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 n-k 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. 45-67.

LINKS

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

FORMULA

T(n,k)=Sum((-1)^j*binom(n-j,j)*binom(n-2j,k)*[c((n-k)/2-2j)*c((n-k)/2-j+2)-c((n-k)/2-j+1)^2], j=0..(n-k)/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`(n-l, 2) = 0 then sum((-1)^b*binomial(n-b, b)*binomial(n-2*b, l)*(c((1/2)*n-(1/2)*l-b)*c((1/2)*n-(1/2)*l-b+2)-c((1/2)*n-(1/2)*l-b+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

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Last modified February 23 08:00 EST 2019. Contains 320420 sequences. (Running on oeis4.)