A377866 Number of subwords of the form DUUD or DDUUD in nondecreasing Dyck paths of length 2n.

0, 0, 0, 1, 5, 18, 59, 185, 564, 1685, 4957, 14406, 41455, 118321, 335400, 945193, 2650229, 7398330, 20573219, 57013865, 157517532, 433993661, 1192779085, 3270835566, 8950887895, 24448816993, 66665369424, 181489721425, 493361278949

Offset: 0

Author

Rigoberto Florez, Nov 10 2024

Keywords

• nonn
• easy

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, 17, 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).

Crossrefs

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

Programs

Mathematica

Table[If[n<3,0,(2*n*LucasL[2*n-5]-6*Fibonacci[2*n-6]-Fibonacci[2*n-7])/5], {n,0,20}]

Formula

a(n) = (2*n*L(2*n-5) - 6*F(2*n-6) - F(2*n-7))/5 for n>=3, where F(n)=A000045(n) and L(n)=A000032(n).

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

E.g.f.: exp(3*x/2)*(5*(35 - 8x)*cosh(sqrt(5)*x/2) - sqrt(5)*(79 - 20*x)*sinh(sqrt(5)*x/2))/25 - 7 - x. - Stefano Spezia, Nov 10 2024