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A187253
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Triangle read by rows: T(n,k) is the number of 3-noncrossing RNA structures on n vertices having k isolated vertices.
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3
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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
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OFFSET
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0,13
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COMMENTS
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Sum of entries in row n is A133365(n).
T(n,k)=0 if n-k is odd.
Sum_{k=0..n} k*T(n,k) = A187255(n).
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LINKS
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FORMULA
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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), where c(n)=A000108(n) are the Catalan numbers (see Corollary 2 in the Jin et al. reference).
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EXAMPLE
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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.
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MAPLE
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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
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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