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 A187255 Number of isolated vertices in all 3-noncrossing RNA structures on n vertices. 1
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS a(n) = Sum_{k=0..n} k*A187253(n,k). 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 FORMULA 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. 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

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Last modified August 7 17:44 EDT 2020. Contains 336278 sequences. (Running on oeis4.)