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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.

%I A210545 #17 Aug 07 2023 08:04:16
%S A210545 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,
%T A210545 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,
%U A210545 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
%N 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.
%H A210545 R. H. Hardin, <a href="/A210545/b210545.txt">Table of n, a(n) for n = 1..9999</a>
%H A210545 Rigoberto Flórez, José L. Ramírez, Fabio A. Velandia, and Diego Villamizar, <a href="https://arxiv.org/abs/2308.02059">Some Connections Between Restricted Dyck Paths, Polyominoes, and Non-Crossing Partitions</a>, arXiv:2308.02059 [math.CO], 2023. See Table 1 p. 13.
%F A210545 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.
%e A210545 Some solutions for n=13 k=4
%e A210545 ..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
%e A210545 ..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
%e A210545 ..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
%e A210545 ..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
%e A210545 ..1....1....1....1....1....1....0....0....1....0....0....1....1....0....1....1
%e A210545 ..0....0....1....1....1....1....1....0....0....0....1....1....0....1....1....0
%e A210545 ..1....0....0....1....1....1....0....1....1....0....0....0....0....1....1....0
%e A210545 ..0....1....1....0....1....1....0....1....0....0....1....0....0....1....1....1
%e A210545 ..1....1....0....0....1....2....1....1....0....1....1....0....0....1....2....1
%e A210545 ..0....1....0....1....0....1....1....1....1....0....1....1....1....1....2....1
%e A210545 ..0....1....0....2....1....0....0....1....1....0....2....0....1....1....2....1
%e A210545 ..1....2....0....1....1....0....1....0....0....1....1....0....0....2....1....0
%e A210545 ..0....2....1....0....2....0....0....0....2....1....2....1....0....0....0....1
%e A210545 Table starts
%e A210545 ..........1.......1......1.....1....1...1...1
%e A210545 ..........2.......1......1.....1....1...1...1
%e A210545 ..........5.......2......1.....1....1...1...1
%e A210545 .........15.......4......2.....1....1...1...1
%e A210545 .........52.......9......4.....2....1...1...1
%e A210545 ........203......23......8.....4....2...1...1
%e A210545 ........877......65.....17.....8....4...2...1
%e A210545 .......4140.....199.....40....16....8...4...2
%e A210545 ......21147.....654....104....33...16...8...4
%e A210545 .....115975....2296....291....73...32..16...8
%e A210545 .....678570....8569....857...177...65..32..16
%e A210545 ....4213597...33825...2634...467..138..64..32
%e A210545 ...27644437..140581...8455..1309..315.129..64
%e A210545 ..190899322..612933..28424..3813..782.267.128
%e A210545 .1382958545.2795182.100117.11409.2090.582.257
%Y A210545 Cf. A000110 (column 1), A007476 (column 2), A210540 (column 3).
%K A210545 nonn,tabl
%O A210545 1,3
%A A210545 _R. H. Hardin_, Mar 22 2012