# [seqfan] Mixture of partitioning and divisibility, formulas?

Ron Hardin rhhardin at att.net
Sun May 20 12:51:11 CEST 2012

```Maybe interesting to somebody

A212536 (diagonal, rows and columns A212531-A212540)
T(n,k)=Number of nondecreasing sequences of n 1..k integers with every element
dividing the sequence sum
Table starts
.1.2..3..4..5...6...7...8...9..10..11...12...13...14...15...16...17...18...19
.1.2..3..4..5...6...7...8...9..10..11...12...13...14...15...16...17...18...19
.1.3..5..7..8..11..12..14..16..18..19...22...23...25...27...29...30...33...34
.1.3..5.10.12..17..18..23..26..30..31...40...41...43...47...52...53...59...60
.1.4..8.15.21..30..33..46..53..66..67...87...88...95..111..125..126..143..144
.1.4..8.15.21..40..44..64..76.103.104..148..149..165..197..229..230..271..272
.1.5.12.24.33..69..83.116.145.188.193..290..293..332..428..496..497..606..607
.1.5.12.29.40..91.106.161.202.266.272..474..478..561..747..874..876.1141.1142
.1.6.16.39.57.130.157.245.331.439.455..867..878.1034.1417.1646.1651.2236.2240
.1.6.16.45.70.166.200.334.451.644.665.1424.1440.1713.2384.2785.2793.3927.3932
Some solutions for n=8 k=4
..1....1....2....1....1....1....1....2....1....1....1....1....2....2....1....1
..1....2....2....2....1....1....1....3....1....1....1....1....2....2....2....1
..2....3....3....2....1....1....1....3....1....2....1....2....2....4....2....2
..2....3....3....2....1....1....1....3....1....2....1....2....2....4....3....2
..2....3....3....2....1....2....2....3....1....2....1....2....4....4....4....3
..4....4....3....3....2....2....2....3....1....2....1....2....4....4....4....3
..4....4....4....3....2....4....2....3....2....2....2....2....4....4....4....3
..4....4....4....3....3....4....2....4....2....2....4....4....4....4....4....3
Column 3 is A000212(floor((n+5)/2))
Row 3 is A106252

The corresponding thing for non-dividing begins (being worked on by a laptop at
the moment)

T(n,k)=Number of nondecreasing sequences of n 1..k integers with no element
dividing the sequence sum
Table starts
.0.0.0..0..0...0...0....0....0....0.....0.....0.....0.....0.....0......0......0
.0.0.1..2..5...7..12...16...22...28....37....43....54....64....75.....86....101
.0.0.1..3..9..16..29...43...64...92...127...168...219...281...355....435....531
.0.0.1..5.15..29..59..103..168..259...386...553...772..1043..1401...1832...2356
.0.0.2..6.22..52.112..212..376..640..1011..1560..2293..3328..4711...6524...8765
.0.0.2..9.32..82.199..407..796.1424..2407..3948..6166..9456.14171..20556..29007
.0.0.2.12.40.122.319..722.1503.2872..5159..9087.15030.24441.38349..58701..86682
.0.0.3.15.59.182.503.1214.2693.5517.10574.19715.34318.58653.96517.154975.238472
All solutions for n=8 k=4
..2....2....2....3....3....2....2....2....2....3....2....2....2....3....2..
..2....3....2....4....3....2....2....2....2....3....2....2....3....3....3..
..2....3....3....4....3....2....3....2....2....3....2....2....3....3....4..
..2....3....3....4....4....2....3....3....2....3....3....2....3....3....4..
..2....3....3....4....4....2....3....3....2....3....4....3....3....3....4..
..3....3....4....4....4....2....3....3....2....3....4....4....3....3....4..
..3....3....4....4....4....2....3....4....3....4....4....4....4....3....4..
..3....3....4....4....4....3....4....4....4....4....4....4....4....4....4..
Row 2 is A161664 (cicada cycles)
Empirical column 4: a(n)=a(n-1)+a(n-3)-a(n-5)-a(n-7)+a(n-8)
Empirical column 5: (order 43 recurrence)

rhhardin at mindspring.com
rhhardin at att.net (either)

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