OFFSET
0,4
COMMENTS
For "secondary structure" and "stack" see the Hofacker et al. reference, p. 209.
LINKS
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
G.f.: G=G(z) satisfies G = 1+zG +fG(G-1)/(1+f), where f = z^2/(1-z^4).
a(n) = A202848(n,0).
D-finite with recurrence (n+2)*a(n) +(-2*n-1)*a(n-1) +(n-1)*a(n-2) +3*(-2*n+5)*a(n-3) +(-n+7)*a(n-6) +3*(2*n-17)*a(n-7) +(-n+10)*a(n-8) +(-2*n+23)*a(n-9) +(n-13)*a(n-10)=0. - R. J. Mathar, Jul 26 2022
EXAMPLE
a(5)=7; 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; only the last one has stacks of even length.
MAPLE
f := z^2/(1-z^4): eq := G = 1+z*G+f*G*(G-1)/(1+f): G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 37)): seq(coeff(Gser, z, n), n = 0 .. 33);
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Dec 26 2011
STATUS
approved