Geodesic
Finite sequences dominated by the squares
Journal of integer sequences, Tome 18 (2015) no. 1
We evaluate the number $u(n)$ of length-$n$ finite sequences $(a_{k})_{1 \le k \le n}$ of natural numbers that satisfy the inequality $a_{k} \le k^{2}$ for all $k$. We thus determine two recurrence relations for $u(n)$ by two different methods, and we give an explicit expression in closed form for it.
Classification : 05A10, 05A15, 05A19, 11B65, 11B75
Keywords: finite sequence, binomial coefficient, combinatorics, enumeration 
@article{JIS_2015__18_1_a0,
     author = {Haddad,  L. and Helou,  C.},
     title = {Finite sequences dominated by the squares},
     journal = {Journal of integer sequences},
     year = {2015},
     volume = {18},
     number = {1},
     zbl = {1310.11027},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2015__18_1_a0/}
}
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Haddad,  L.; Helou,  C. Finite sequences dominated by the squares. Journal of integer sequences, Tome 18 (2015) no. 1. http://geodesic.mathdoc.fr/item/JIS_2015__18_1_a0/