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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 (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 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(n-2)=A171854(n) (n>=2).

REFERENCES

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.

LINKS

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

FORMULA

G.f.: g(z) = z^2*(1-z^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(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 odd length, respectively.

MAPLE

g := z^2*(1-z^2)*S*(S-1)/((1+z^2)*(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. 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

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Last modified December 13 03:41 EST 2019. Contains 329968 sequences. (Running on oeis4.)