

A202846


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


3



0, 0, 0, 1, 3, 6, 16, 44, 113, 290, 749, 1930, 4966, 12776, 32870, 84577, 217665, 560328, 1442893, 3716837, 9577805, 24689612, 63667585, 164239124, 423824628, 1094065998, 2825169786, 7297681867, 18856458451, 48737762624, 126007604078, 325873570924, 842982118807
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OFFSET

0,5


COMMENTS

For "secondary structure" and "stack" see the Hofacker et al. reference, p. 209.
Number of stacks of even length in all 2ndary structures of size n+2.
a(n) = Sum(k*A202845(n,k), k>=0).
a(n) = Sum(k*A202848(n+2,k), k>=0).
a(n)+a(n2)=A171854(n) (n>=2).


REFERENCES

I. L. Hofacker, P. Schuster and P. F. Stadler, Combinatorics of RNA secondary structures, Discrete Appl. Math., 88, 1998, 207237.
P. R. Stein and M. S. Waterman, On some new sequences generalizing the Catalan and Motzkin numbers, Discrete Math., 26, 1979, 261272.


LINKS

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


FORMULA

G.f.: g(z) = z^2*(1z^2)^2*S(S  1)/[(1+z^2)(1  z + z^2 2*z^2*S)], where S is defined by S = 1 + z*S + z^2*S(S1) (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 odd length, respectively.


MAPLE

g := z^2*(1z^2)*S*(S1)/((1+z^2)*(1z+z^22*z^2*S)): S := ((1z+z^2sqrt(12*zz^22*z^3+z^4))*1/2)/z^2: gser := series(g, z = 0, 35): seq(coeff(gser, z, n), n = 0 .. 32);


CROSSREFS

Cf. A004148, A171854, A202845, A023427, A202848, A202849
Sequence in context: A007561 A274295 A192676 * A107269 A086811 A106361
Adjacent sequences: A202843 A202844 A202845 * A202847 A202848 A202849


KEYWORD

nonn


AUTHOR

Emeric Deutsch, Dec 26 2011


STATUS

approved



