A202846 Number of stacks of odd length in all 2ndary structures of size n.

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

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.

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*A202845(n,k), k>=0).

a(n) = Sum(k*A202848(n+2,k), k>=0).

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

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).

Conjecture D-finite with recurrence +(n+2)*(13230*n^2-96611*n+147133)*a(n) +(-44206*n^3+292903*n^2-261197*n-341332)*a(n-1) +2*(17746*n^3-141629*n^2+231187*n+123600)*a(n-2) +2*(-26460*n^3+157889*n^2-64195*n-381418)*a(n-3) +2*(35492*n^3-320849*n^2+745453*n-240088)*a(n-4) +2*(-13230*n^3+98869*n^2-160610*n-79637)*a(n-5) +(48722*n^3-428591*n^2+982443*n-433110)*a(n-6) -(n-6)*(17746*n^2-68387*n+43705)*a(n-7)=0. - R. J. Mathar, Jul 26 2022

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 A333026 A086811

Adjacent sequences: A202843 A202844 A202845 * A202847 A202848 A202849

Keyword

nonn

Author

Emeric Deutsch, Dec 26 2011

Status

approved