VOOZH about

URL: https://oeis.org/A375995

⇱ A375995 - OEIS

login
A375995
Number of subwords of the form UUUU in nondecreasing Dyck paths of length 2n.
6
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
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.
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
KEYWORD
nonn,easy
AUTHOR
Rigoberto Florez, Nov 03 2024
STATUS
approved