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 A377866 #17 Mar 04 2025 08:36:13
%S A377866 0,0,0,1,5,18,59,185,564,1685,4957,14406,41455,118321,335400,945193,
%T A377866 2650229,7398330,20573219,57013865,157517532,433993661,1192779085,
%U A377866 3270835566,8950887895,24448816993,66665369424,181489721425,493361278949
%N A377866 Number of subwords of the form DUUD or DDUUD in nondecreasing Dyck paths of length 2n.
%C A377866 A Dyck path is nondecreasing if the y-coordinates of its valleys form a nondecreasing sequence.
%H A377866 E. Barcucci, A. Del Lungo, S. Fezzi, and R. Pinzani, <a href="http://dx.doi.org/10.1016/S0012-365X(97)82778-1">Nondecreasing Dyck paths and q-Fibonacci numbers</a>, Discrete Math., 170 (1997), 211-217.
%H A377866 Éva Czabarka, Rigoberto Flórez, Leandro Junes and José L. Ramírez, <a href="https://doi.org/10.1016/j.disc.2018.06.032">Enumerations of peaks and valleys on non-decreasing Dyck paths</a>, Discrete Math., Vol. 341, No. 10 (2018), pp. 2789-2807. See p. 2798.
%H A377866 Rigoberto Flórez, Leandro Junes, Luisa M. Montoya, and José L. Ramírez, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL28/Florez/florez51.html">Counting Subwords in Non-Decreasing Dyck Paths</a>, J. Int. Seq. (2025) Vol. 28, Art. No. 25.1.6. See pp. 15, 17, 19.
%H A377866 Rigoberto Flórez, Leandro Junes, and José L. Ramírez, <a href="https://doi.org/10.1016/j.disc.2019.06.018">Enumerating several aspects of non-decreasing Dyck paths</a>, Discrete Mathematics, Vol. 342, Issue 11 (2019), 3079-3097. See page 3092.
%H A377866 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (6,-11,6,-1).
%F A377866 a(n) = (2*n*L(2*n-5) - 6*F(2*n-6) - F(2*n-7))/5 for n>=3, where F(n)=A000045(n) and L(n)=A000032(n).
%F A377866 G.f.: -x^3*(x^2+x-1)/ (x^2-3*x+1)^2.
%F A377866 E.g.f.: exp(3*x/2)*(5*(35 - 8x)*cosh(sqrt(5)*x/2) - sqrt(5)*(79 - 20*x)*sinh(sqrt(5)*x/2))/25 - 7 - x. - _Stefano Spezia_, Nov 10 2024
%t A377866 Table[If[n<3,0,(2*n*LucasL[2*n-5]-6*Fibonacci[2*n-6]-Fibonacci[2*n-7])/5], {n,0,20}]
%Y A377866 Cf. A000032, A000045, A377679, A377670, A375995.
%K A377866 nonn,easy
%O A377866 0,5
%A A377866 _Rigoberto Florez_, Nov 10 2024