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 A378383 #17 Mar 04 2025 08:36:16
%S A378383 0,0,0,1,5,19,64,202,612,1803,5206,14809,41650,116114,321478,885169,
%T A378383 2426462,6627499,18048088,49026874,132901176,359625015,971639014,
%U A378383 2621683741,7065545950,19022080034,51163908874,137499581917,369235213742,990822728623,2657069356996
%N A378383 Number of subwords of the form UDDD in nondecreasing Dyck paths of length 2*n.
%C A378383 A Dyck path is nondecreasing if the y-coordinates of its valleys form a nondecreasing sequence.
%H A378383 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 A378383 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 A378383 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 A378383 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 A378383 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (10,-39,74,-69,28,-4).
%F A378383 a(n) =((n-3)*L(2n-5)+L(2n-3)+F(2n+2) -5*(n+5)*2^(n-4))/5 for n>=3, where F(n) = A000045(n) and L(n) = A000032(n).
%F A378383 G.f.: (-x^5+2 x^4-5 x^3+8 x^2-5 x+1)*x^3/(2 x^3-7 x^2+5 x-1)^2.
%t A378383 Table[If[n < 3, 0, (1/5)((n-3)LucasL[2n-5]+LucasL[2n-3]+Fibonacci[2n+2]-5(n+5) 2^(n-4))], {n,0,26}]
%Y A378383 Cf. A000032, A000045, A377679, A377670, A375995, A377857, A377866, A377867.
%K A378383 nonn,easy
%O A378383 0,5
%A A378383 _Rigoberto Florez_, Nov 24 2024