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%I #13 Jul 26 2022 14:50:06
%S 1,1,1,2,4,7,14,31,66,141,313,702,1577,3581,8207,18903,43770,101903,
%T 238282,559322,1317717,3114676,7383914,17552857,41831618,99923471,
%U 239200459,573750288,1378763083,3319005743,8002573350,19324601494,46731582653,113160019865
%N Number of secondary structures of size n having no stacks of even length.
%C For "secondary structure" and "stack" see the Hofacker et al. reference, p. 209.
%H 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.
%H 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.
%F G.f.: G=G(z) satisfies G = 1+zG +fG(G-1)/(1+f), where f = z^2/(1-z^4).
%F a(n) = A202848(n,0).
%F 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
%e 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.
%p 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);
%Y Cf. A202845, A202846, A023427, A202848
%K nonn
%O 0,4
%A _Emeric Deutsch_, Dec 26 2011
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