A377867 Number of subwords of the form DDDD in nondecreasing Dyck paths of length 2n.

0, 0, 0, 0, 1, 7, 33, 131, 473, 1608, 5242, 16567, 51123, 154793, 461525, 1358646, 3957088, 11420995, 32707809, 93040751, 263113505, 740238852, 2073098086, 5782387855, 16070206191, 44516728277, 122956408493, 338707969266, 930787894348, 2552224341403, 6984100641117

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 (10,-39,74,-69,28,-4).

Formula

a(n) = (3*(n-2)*L(2*n-4) - 3*F(2*n+1))/5 + (n+9)*2^(n-4) for n>=3, where F(n) = A000045(n) and L(n) = A000032(n).

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

Mathematica

Table[If[n < 3, 0, (3*(n-2)*LucasL[2*n-4]-3*Fibonacci[2*n+1])/5+(n+9)*2^(n-4)], {n, 0, 20}]

Crossrefs

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

Sequence in context: A320907 A258458 A320546 * A066810 A262600 A034577

Adjacent sequences: A377864 A377865 A377866 * A377868 A377869 A377870

Keyword

nonn,easy,changed

Author

Rigoberto Florez, Nov 10 2024

Status

approved