A377679 - OEIS

%I #19 Nov 11 2024 07:48:16

%S 0,0,0,1,6,26,97,333,1085,3411,10448,31376,92773,270907,783003,

%T 2243815,6383550,18048494,50755897,142067625,396014681,1099863867,

%U 3044737100,8404071596,23135752141,63538808311,174120317367,476207551183

%N Number of subwords of the form DDD 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, 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_05">Index entries for linear recurrences with constant coefficients</a>, signature (8, -23, 28, -13, 2).

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

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

%t Table[If[n<2,0,n Fibonacci[2 n-3]-LucalL[2 n-2]+2^(n-2)],{n,0,20}]

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

%K nonn,easy

%O 0,5

%A _Rigoberto Florez_, Nov 03 2024