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A211700 T(n,k)=Number of nonnegative integer arrays of length n+2k-2 with new values introduced in order 0 upwards and every value appearing only in runs of at least k.

%I A211700 #14 Jul 22 2025 22:48:49
%S A211700 1,1,2,1,2,5,1,2,3,15,1,2,3,6,52,1,2,3,4,11,203,1,2,3,4,7,23,877,1,2,
%T A211700 3,4,5,12,47,4140,1,2,3,4,5,8,19,103,21147,1,2,3,4,5,6,13,33,226,
%U A211700 115975,1,2,3,4,5,6,9,20,59,518,678570,1,2,3,4,5,6,7,14,29,102,1200,4213597,1,2,3,4
%N A211700 T(n,k)=Number of nonnegative integer arrays of length n+2k-2 with new values introduced in order 0 upwards and every value appearing only in runs of at least k.
%H A211700 R. H. Hardin, <a href="/A211700/b211700.txt">Table of n, a(n) for n = 1..9999</a>
%H A211700 Beáta Bényi, Toufik Mansour, and José L. Ramírez, <a href="https://ajc.maths.uq.edu.au/pdf/84/ajc_v84_p325.pdf">Set partitions and non-crossing partitions with l-neighbors and l-isolated elements</a>, Australasian J. Comb. (2022) Vol. 84, No. 2, 325-340.
%e A211700 Table starts
%e A211700 ..........1.....1....1...1...1...1..1..1..1..1..1..1..1..1
%e A211700 ..........2.....2....2...2...2...2..2..2..2..2..2..2..2..2
%e A211700 ..........5.....3....3...3...3...3..3..3..3..3..3..3..3..3
%e A211700 .........15.....6....4...4...4...4..4..4..4..4..4..4..4..4
%e A211700 .........52....11....7...5...5...5..5..5..5..5..5..5..5..5
%e A211700 ........203....23...12...8...6...6..6..6..6..6..6..6..6..6
%e A211700 ........877....47...19..13...9...7..7..7..7..7..7..7..7..7
%e A211700 .......4140...103...33..20..14..10..8..8..8..8..8..8..8..8
%e A211700 ......21147...226...59..29..21..15.11..9..9..9..9..9..9..9
%e A211700 .....115975...518..102..45..30..22.16.12.10.10.10.10.10.10
%e A211700 .....678570..1200..182..73..41..31.23.17.13.11.11.11.11.11
%e A211700 ....4213597..2867..334.118..59..42.32.24.18.14.12.12.12.12
%e A211700 ...27644437..6946..608.185..89..55.43.33.25.19.15.13.13.13
%e A211700 ..190899322.17234.1121.294.136..75.56.44.34.26.20.16.14.14
%e A211700 .1382958545.43393.2109.480.205.107.71.57.45.35.27.21.17.15
%e A211700 All solutions for n=5 k=4
%e A211700 ..0....0....0....0....0
%e A211700 ..0....0....0....0....0
%e A211700 ..0....0....0....0....0
%e A211700 ..0....0....0....0....0
%e A211700 ..1....0....0....0....0
%e A211700 ..1....0....0....0....1
%e A211700 ..1....0....1....0....1
%e A211700 ..1....0....1....1....1
%e A211700 ..1....0....1....1....1
%e A211700 ..1....0....1....1....1
%e A211700 ..1....0....1....1....1
%Y A211700 Cf. A000110 (column 1), A211694 (column 2), A211695 (column 3).
%K A211700 nonn,tabl
%O A211700 1,3
%A A211700 _R. H. Hardin_, Apr 19 2012