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D-finite with recurrence +(n+4)*a(n) +(-2*n-5)*a(n-1) +(-n-1)*a(n-2) +2*(2*n-1)*a(n-3) +(-n+2)*a(n-4) +4*(-2*n+7)*a(n-5) +3*(n-5)*a(n-6) +3*(2*n-13)*a(n-7) +2*(-n+8)*a(n-8) +2*(-2*n+19)*a(n-9) +(n-11)*a(n-10)=0. - R. J. Mathar, Jul 26 2022

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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.

I. L. Hofacker, P. Schuster and P. F. Stadler, <a href="https://doi.org/10.1016/S0166-218X(98)00073-0">Combinatorics of RNA secondary structures</a>, Discrete Appl. Math., 88, 1998, 207-237.

P. R. Stein and M. S. Waterman, <a href="https://doi.org/10.1016/0012-365X(79)90033-5">On some new sequences generalizing the Catalan and Motzkin numbers</a>, Discrete Math., 26 (1979), 261-272.

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_Emeric Deutsch (deutsch(AT)duke.poly.edu), _, Dec 25 2011

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allocated for Emeric DeutschNumber of secondary structures of size n having no stacks of length 1.

1, 1, 1, 1, 1, 2, 4, 8, 14, 23, 38, 65, 117, 214, 391, 708, 1278, 2318, 4238, 7803, 14419, 26684, 49433, 91736, 170656, 318280, 594905, 1113868, 2088554, 3921505, 7373367, 13883045, 26174600, 49408932, 93372078, 176637791, 334491586, 634023965, 1202894908, 2284187117

0,6

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

a(n) = A202838(n,0).

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.

G.f. G=G(z) satisfies G = 1+zG +fG(G-1)/(1+f), where f = z^4/(1-z^2).

a(5)=2; 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.

f := z^4/(1-z^2): eq := G = 1+z*G+f*G*(G-1)/(1+f): G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 42)): seq(coeff(Gser, z, n), n = 0 .. 39);

Cf. A202838, A202839, A202841, A202842, A202843, A202844

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Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 25 2011

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