A210545 T(n,k) = number of arrays of n nonnegative integers with value i>0 appearing only after i-1 has appeared at least k times.

1, 1, 2, 1, 1, 5, 1, 1, 2, 15, 1, 1, 1, 4, 52, 1, 1, 1, 2, 9, 203, 1, 1, 1, 1, 4, 23, 877, 1, 1, 1, 1, 2, 8, 65, 4140, 1, 1, 1, 1, 1, 4, 17, 199, 21147, 1, 1, 1, 1, 1, 2, 8, 40, 654, 115975, 1, 1, 1, 1, 1, 1, 4, 16, 104, 2296, 678570, 1, 1, 1, 1, 1, 1, 2, 8, 33, 291, 8569, 4213597, 1, 1, 1, 1

Offset

1,3

Links

R. H. Hardin, Table of n, a(n) for n = 1..9999

Rigoberto Flórez, José L. Ramírez, Fabio A. Velandia, and Diego Villamizar, Some Connections Between Restricted Dyck Paths, Polyominoes, and Non-Crossing Partitions, arXiv:2308.02059 [math.CO], 2023. See Table 1 p. 13.

Formula

T(n,k)=1 if n<=k else Sum_{i=0..n-k} binomial(n-k,i)*T(i,k). Proved by R. J. Mathar in the Sequence Fans Mailing List.

Example

Some solutions for n=13 k=4
..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
..1....1....1....1....1....1....0....0....1....0....0....1....1....0....1....1
..0....0....1....1....1....1....1....0....0....0....1....1....0....1....1....0
..1....0....0....1....1....1....0....1....1....0....0....0....0....1....1....0
..0....1....1....0....1....1....0....1....0....0....1....0....0....1....1....1
..1....1....0....0....1....2....1....1....0....1....1....0....0....1....2....1
..0....1....0....1....0....1....1....1....1....0....1....1....1....1....2....1
..0....1....0....2....1....0....0....1....1....0....2....0....1....1....2....1
..1....2....0....1....1....0....1....0....0....1....1....0....0....2....1....0
..0....2....1....0....2....0....0....0....2....1....2....1....0....0....0....1
Table starts
..........1.......1......1.....1....1...1...1
..........2.......1......1.....1....1...1...1
..........5.......2......1.....1....1...1...1
.........15.......4......2.....1....1...1...1
.........52.......9......4.....2....1...1...1
........203......23......8.....4....2...1...1
........877......65.....17.....8....4...2...1
.......4140.....199.....40....16....8...4...2
......21147.....654....104....33...16...8...4
.....115975....2296....291....73...32..16...8
.....678570....8569....857...177...65..32..16
....4213597...33825...2634...467..138..64..32
...27644437..140581...8455..1309..315.129..64
..190899322..612933..28424..3813..782.267.128
.1382958545.2795182.100117.11409.2090.582.257

Crossrefs

Cf. A000110 (column 1), A007476 (column 2), A210540 (column 3).

Sequence in context: A169950 A088347 A069568 * A141323 A210876 A174785

Adjacent sequences: A210542 A210543 A210544 * A210546 A210547 A210548

Keyword

nonn,tabl

Author

R. H. Hardin, Mar 22 2012

Status

approved