A377857 - OEIS

%I #22 Mar 04 2025 08:35:56

%S 0,0,0,1,5,18,60,191,589,1775,5257,15360,44394,127171,361595,1021693,

%T 2871245,8031246,22372344,62096135,171797257,473928875,1304007889,

%U 3579517116,9804791910,26804181643,73145473655,199276078201,542076556949,1472491141770,3994615719732

%N Number of subwords of the form UUUD in nondecreasing Dyck paths of length 2n.

%C A Dyck path is nondecreasing if the y-coordinates of its valleys form a nondecreasing sequence.

%H 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 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 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 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 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (6,-11,6,-1).

%F a(n) = n*F(2*n-5) - L(2*n-6) for n>=3, where F(n) = A000045(n) and L(n) = A000032(n).

%F G.f.: x^3*(1 - x)^2*(1 + x)/(1 - 3*x + x^2)^2.

%F a(n) = A317408(n-2)-A317408(n-3) = A030267(n-2)+A030267(n-3). - _R. J. Mathar_, Dec 16 2024

%t Table[If[n<3,0,n Fibonacci[2n-5]-LucasL[2n-6]], {n,0,30}]

%Y Cf. A000032, A000045, A377679,  A377670, A375995.

%K nonn,easy,changed

%O 0,5

%A _Rigoberto Florez_, Nov 09 2024