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A377670
Number of subwords of the form UDD in nondecreasing Dyck paths of length 2n.
6
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
0,4
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.
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
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}]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Rigoberto Florez, Nov 03 2024
STATUS
approved