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 A377867 #13 Mar 04 2025 08:36:24
%S A377867 0,0,0,0,1,7,33,131,473,1608,5242,16567,51123,154793,461525,1358646,
%T A377867 3957088,11420995,32707809,93040751,263113505,740238852,2073098086,
%U A377867 5782387855,16070206191,44516728277,122956408493,338707969266,930787894348,2552224341403,6984100641117
%N A377867 Number of subwords of the form DDDD in nondecreasing Dyck paths of length 2n.
%C A377867 A Dyck path is nondecreasing if the y-coordinates of its valleys form a nondecreasing sequence.
%H A377867 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 A377867 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 A377867 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, 19.
%H A377867 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 A377867 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (10,-39,74,-69,28,-4).
%F A377867 a(n) = (3*(n-2)*L(2*n-4) - 3*F(2*n+1))/5 + (n+9)*2^(n-4) for n>=3, where F(n) = A000045(n) and L(n) = A000032(n).
%F A377867 G.f.: x^4*(1 - 3*x + 2*x^2 + x^4)/((1 - 2*x)^2*(1 - 3*x + x^2)^2).
%t A377867 Table[If[n < 3, 0, (3*(n-2)*LucasL[2*n-4]-3*Fibonacci[2*n+1])/5+(n+9)*2^(n-4)], {n,0,20}]
%Y A377867 Cf. A000032, A000045, A377679, A377670, A375995.
%K A377867 nonn,easy
%O A377867 0,6
%A A377867 _Rigoberto Florez_, Nov 10 2024