A377857 Number of subwords of the form UUUD in nondecreasing Dyck paths of length 2n.

0, 0, 0, 1, 5, 18, 60, 191, 589, 1775, 5257, 15360, 44394, 127171, 361595, 1021693, 2871245, 8031246, 22372344, 62096135, 171797257, 473928875, 1304007889, 3579517116, 9804791910, 26804181643, 73145473655, 199276078201, 542076556949, 1472491141770, 3994615719732

Offset

0,5

Comments

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

Links

Table of n, a(n) for n=0..30.

E. Barcucci, A. Del Lungo, S. Fezzi, and R. Pinzani, Nondecreasing Dyck paths and q-Fibonacci numbers, Discrete Math., 170 (1997), 211-217.

Éva Czabarka, Rigoberto Flórez, Leandro Junes and José L. Ramírez, Enumerations of peaks and valleys on non-decreasing Dyck paths, Discrete Math., Vol. 341, No. 10 (2018), pp. 2789-2807. See p. 2798.

Rigoberto Flórez, Leandro Junes, and José L. Ramírez, Enumerating several aspects of non-decreasing Dyck paths, Discrete Mathematics, Vol. 342, Issue 11 (2019), 3079-3097. See page 3092.

Index entries for linear recurrences with constant coefficients, signature (6,-11,6,-1).

Formula

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

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

Mathematica

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

Crossrefs

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

Sequence in context: A377866 A284968 A222567 * A099449 A104630 A062809

Adjacent sequences: A377854 A377855 A377856 * A377858 A377859 A377860

Keyword

nonn,easy,new

Author

Rigoberto Florez, Nov 09 2024

Status

approved