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 A377670 #42 Mar 05 2025 02:06:24
%S A377670 0,0,1,4,14,45,138,411,1200,3454,9836,27779,77938,217493,604222,
%T A377670 1672246,4613030,12689265,34817418,95320335,260436588,710278318,
%U A377670 1933906496,5257545599,14273273314,38699274665,104799960058,283487736166,766045036730,2067997219629,5577597593466,15030365074659,40470488092008
%N A377670 Number of subwords of the form UDD in nondecreasing Dyck paths of length 2n.
%C A377670 A Dyck path is nondecreasing if the y-coordinates of its valleys form a nondecreasing sequence.
%C A377670 a(n) also represents the number of subwords of the form UUDDD in nondecreasing Dyck paths of length 2n.
%H A377670 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 A377670 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 A377670 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 A377670 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. 6, 17, 19.
%H A377670 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (8,-23,28,-13,2).
%F A377670 a(n) = (2*F(2*n-3) + n*L(2*n-3) + 2*L(2*n-2) - 5*2^(n-2))/5 for n>=2, where F(n) = A000045(n) and L(n) = A000032(n).
%F A377670 G.f.: x^2*(1-x)*(x^3-2*x^2+3*x-1)/((2*x-1)*(x^2-3*x+1)^2). - _Alois P. Heinz_, Nov 03 2024
%F A377670 E.g.f.: (4*exp(3*x/2)*(5*(10 - x)*cosh(sqrt(5)*x/2) - sqrt(5)*(18 - 5*x)*sinh(sqrt(5)*x/2)) - 25*(7 + exp(2*x) + 2*x))/100. - _Stefano Spezia_, Mar 04 2025
%t A377670 Table[If[n<2, 0,(2*Fibonacci[2n-3] + n*LucasL[2n-3]+2 LucasL[2n-2]-5*2^(n-2))/5], {n,0,20}]
%Y A377670 Cf. A000032, A000045, A377679, A375995.
%K A377670 nonn,easy
%O A377670 0,4
%A A377670 _Rigoberto Florez_, Nov 03 2024