A377679 Number of subwords of the form DDD in nondecreasing Dyck paths of length 2n.

0, 0, 0, 1, 6, 26, 97, 333, 1085, 3411, 10448, 31376, 92773, 270907, 783003, 2243815, 6383550, 18048494, 50755897, 142067625, 396014681, 1099863867, 3044737100, 8404071596, 23135752141, 63538808311, 174120317367, 476207551183

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..27.

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 (8, -23, 28, -13, 2).

Formula

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).

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

Mathematica

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

Crossrefs

Cf. A000032, A000045, A377670, A375995.

Sequence in context: A143132 A055589 A318947 * A320816 A239179 A307309

Adjacent sequences: A377676 A377677 A377678 * A377680 A377681 A377682

Keyword

nonn,easy

Author

Rigoberto Florez, Nov 03 2024

Status

approved