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

0,5

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..28.

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 (6,-11,6,-1).

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

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}]

Crossrefs

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

Sequence in context: A301880 A034567 A133648 * A284968 A222567 A377857

Adjacent sequences: A377863 A377864 A377865 * A377867 A377868 A377869

Keyword

nonn,easy

Author

Rigoberto Florez, Nov 10 2024

Status

approved