A375995 Number of subwords of the form UUUU in nondecreasing Dyck paths of length 2n.

0, 0, 0, 0, 1, 7, 30, 109, 365, 1164, 3593, 10835, 32106, 93845, 271321, 777432, 2211025, 6248479, 17562870, 49132669, 136884293, 379975140, 1051356761, 2900587115, 7981564866, 21911096357, 60021530545, 164095925424, 447823729825, 1220105286199, 3319124711118

Offset

0,6

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, Luisa M. Montoya, and José L. Ramírez, Counting Subwords in Non-Decreasing Dyck Paths, J. Int. Seq. (2025) Vol. 28, Art. No. 25.1.6. See pp. 15, 19.

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) = (2(n-3)*L(2 n-5)-3F(2n-6))/5 for n>=3 and a(n) = 0 for n<=2, F(.) is a Fibonacci number, L(.) is a Lucas number.

G.f.: x^4*(-x^2+x+1)/(x^2-3x+1)^2.

Mathematica

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

Crossrefs

Cf. A000032, A000045, A377679, A377670.

Sequence in context: A364655 A368528 A085277 * A269084 A055269 A026631

Adjacent sequences: A375992 A375993 A375994 * A375996 A375997 A375998

Keyword

nonn,easy,changed

Author

Rigoberto Florez, Nov 03 2024

Status

approved