This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A202846 #14 Jul 26 2022 13:11:21 %S A202846 0,0,0,1,3,6,16,44,113,290,749,1930,4966,12776,32870,84577,217665, %T A202846 560328,1442893,3716837,9577805,24689612,63667585,164239124,423824628, %U A202846 1094065998,2825169786,7297681867,18856458451,48737762624,126007604078,325873570924,842982118807 %N A202846 Number of stacks of odd length in all 2ndary structures of size n. %C A202846 For "secondary structure" and "stack" see the Hofacker et al. reference, p. 209. %C A202846 Number of stacks of even length in all 2ndary structures of size n+2. %H A202846 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 A202846 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 A202846 a(n) = Sum(k*A202845(n,k), k>=0). %F A202846 a(n) = Sum(k*A202848(n+2,k), k>=0). %F A202846 a(n)+a(n-2) = A171854(n) (n>=2). %F A202846 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). %F A202846 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 %e A202846 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. %p A202846 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); %Y A202846 Cf. A004148, A171854, A202845, A023427, A202848, A202849 %K A202846 nonn %O A202846 0,5 %A A202846 _Emeric Deutsch_, Dec 26 2011