A377670 Number of subwords of the form UDD in nondecreasing Dyck paths of length 2n.

0, 0, 1, 4, 14, 45, 138, 411, 1200, 3454, 9836, 27779, 77938, 217493, 604222, 1672246, 4613030, 12689265, 34817418, 95320335, 260436588, 710278318, 1933906496, 5257545599, 14273273314, 38699274665, 104799960058, 283487736166, 766045036730, 2067997219629, 5577597593466, 15030365074659, 40470488092008

Offset: 0

Author

Rigoberto Florez, Nov 03 2024

Keywords

• nonn
• easy

Comments

A Dyck path is nondecreasing if the y-coordinates of its valleys form a nondecreasing sequence.

a(n) also represents the number of subwords of the form UUDDD in nondecreasing Dyck paths of length 2n.

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, and José L. Ramírez, Enumerating several aspects of non-decreasing Dyck paths, Discrete Mathematics, Vol. 342, Issue 11 (2019), 3079-3097. See page 3092.
• 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. 6, 17, 19.
• Index entries for linear recurrences with constant coefficients, signature (8,-23,28,-13,2).

Crossrefs

Cf. A000032, A000045, A377679, A375995.

Programs

Mathematica

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

Formula

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

G.f.: x^2*(1-x)*(x^3-2*x^2+3*x-1)/((2*x-1)*(x^2-3*x+1)^2). - Alois P. Heinz, Nov 03 2024

E.g.f.: (4*exp(3*x/2)*(5*(10 - x)*cosh(sqrt(5)*x/2) - sqrt(5)*(18 - 5*x)*sinh(sqrt(5)*x/2)) - 25*(7 + exp(2*x) + 2*x))/100. - Stefano Spezia, Mar 04 2025