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A202839 Number of stacks of length 1 in all 2ndary structures of size n. 6
0, 0, 0, 1, 3, 6, 16, 43, 110, 284, 733, 1886, 4853, 12486, 32121, 82647, 212699, 547552, 1410023, 3632260, 9360140, 24129284, 62224692, 160522287, 414246823, 1069376386, 2761502201, 7133442743, 18432633823, 47643696626, 123182434292, 318575889057, 824125660356 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

For "secondary structure" and "stack" see the Hofacker et al. reference, p. 209.

Number of stacks of length 2 in all 2ndary structures of size n+2.

Number of stacks of length 3 in all 2ndary structures of size n+4.

LINKS

Table of n, a(n) for n=0..32.

I. L. Hofacker, P. Schuster and P. F. Stadler, Combinatorics of RNA secondary structures, Discrete Appl. Math., 88, 1998, 207-237.

P. R. Stein and M. S. Waterman, On some new sequences generalizing the Catalan and Motzkin numbers, Discrete Math., 26 (1979), 261-272.

FORMULA

a(n) = Sum(k*A202838(n,k), k>=0).

a(n) = Sum(k*A202841(n+2,k), k>=0).

a(n) = Sum(k*A202843(n+4,k), k>=0).

G.f.: g(z) = z^2*(1-z^2)^2*S(S - 1)/(1 - z + z^2 -2*z^2*S), where S is defined by S = 1 + z*S + z^2*S(S-1) (the g.f. of the secondary structure numbers A004148).

EXAMPLE

a(5)=6: representing unpaired vertices by v and arcs by AA, BB, etc., the 8 (= A004148(5)) secondary structures of size 5 are vvvvv, AvAvv, vvAvA, AvvAv, vAvvA, AvvvA, vAvAv, ABvBA; they have 0,1,1,1,1,1,1,0 stacks of length 1, respectively.

MAPLE

g := z^2*(1-z^2)^2*S*(S-1)/(1-z+z^2-2*z^2*S): S := ((1-z+z^2-sqrt(1-2*z-z^2-2*z^3+z^4))*1/2)/z^2: gser := series(g, z = 0, 35): seq(coeff(gser, z, n), n = 0 .. 32);

CROSSREFS

Cf. A202838, A202840, A202841, A202842, A202843, A202844

Sequence in context: A218207 A019497 A091488 * A007561 A274295 A192676

Adjacent sequences:  A202836 A202837 A202838 * A202840 A202841 A202842

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Dec 25 2011

STATUS

approved

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Last modified March 28 07:59 EDT 2020. Contains 333079 sequences. (Running on oeis4.)