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 A110238 #15 Nov 16 2019 20:07:51 %S A110238 1,3,7,1,17,5,41,16,1,98,46,7,235,127,29,1,565,340,99,9,1362,893,310, %T A110238 46,1,3294,2318,921,184,11,7992,5968,2640,650,67,1,19450,15279,7382, %U A110238 2131,309,13,47475,38965,20274,6641,1223,92,1,116204,99101,54935,19958,4404 %N A110238 Triangle read by rows: T(n,k) (0 <= k <= ceiling(n/2)-2) is the number of (1,1) steps starting at level k in all peakless Motzkin paths of length n (can be easily translated into RNA secondary structure terminology). %C A110238 Row n has ceiling(n/2)-1 terms. Row sums yield A110239. %H A110238 W. R. Schmitt and M. S. Waterman, <a href="http://dx.doi.org/10.1016/0166-218X(92)00038-N">Linear trees and RNA secondary structure</a>, Discrete Appl. Math., 51, 317-323, 1994. %H A110238 P. R. Stein and M. S. Waterman, <a href="http://dx.doi.org/10.1016/0012-365X(79)90033-5">On some new sequences generalizing the Catalan and Motzkin numbers</a>, Discrete Math., 26 (1978), 261-272. %H A110238 M. Vauchassade de Chaumont and G. Viennot, <a href="https://www.emis.de/journals/SLC/opapers/s08viennot.html">Polynômes orthogonaux et problèmes d'énumeration en biologie moléculaire</a>, Publ. I.R.M.A. Strasbourg, 1984, 229/S-08, Actes 8e Sem. Lotharingien, pp. 79-86. %F A110238 G.f.: z^2*g^2*(g-1)/(1-tz^2*g^2), where g = 1 + zg + z^2*g(g-1) = (1 - z + z^2 - sqrt(1 - 2z - z^2 - 2z^3 + z^4))/(2z^2) is the g.f. of the RNA secondary structure numbers (A004148). %e A110238 T(5,1)=1 because in the 8 (=A004148(5)) peakless Motzkin paths of length 5, namely HHHHH, UHDHH, UHHDH, UHHHD, HUHDH, HUHHD, HHUHD and U(U)HDD (where U=(1,1), H=(1,0) and D=(1,-1)), we have altogether 1 U step starting at level 1 (shown between parentheses). %e A110238 Triangle starts: %e A110238 1; %e A110238 3; %e A110238 7, 1; %e A110238 17, 5; %e A110238 41, 16, 1; %e A110238 98, 46, 7; %p A110238 g:=(1-z+z^2-sqrt(1-2*z-z^2-2*z^3+z^4))/2/z^2: G:=z^2*g^2*(g-1)/(1-t*z^2*g^2): Gser:=simplify(series(G,z=0,21)): for n from 1 to 17 do P[n]:=coeff(Gser,z^n) od: for n from 3 to 17 do seq(coeff(t*P[n],t^k),k=1..ceil(n/2)-1) od; %Y A110238 Cf. A004148, A110239, A110237. %K A110238 nonn,tabf %O A110238 3,2 %A A110238 _Emeric Deutsch_, Jul 17 2005