# [seqfan] Prove formula?

Ron Hardin rhhardin at att.net
Tue Mar 20 22:36:09 CET 2012

```A formula for the below is easily guessed from the first two columns, where OEIS
gives the formula.

Is it obvious, or easy to prove?  If so "Empirical" can be omitted.

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

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

Some.solutions.for.n=4.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....0....1....1....1....1....1....1....1....1....1....1..
..1....1....0....0....0....0....1....1....0....1....1....0....0....1....1....1..
..1....1....0....0....1....1....1....1....0....1....1....1....1....1....1....0..
..0....1....1....0....0....1....0....0....1....1....0....1....0....1....1....1..
..1....2....1....1....0....0....1....1....1....0....1....1....1....1....0....1..
..2....2....0....0....1....1....1....2....0....0....1....1....0....2....1....1..
..2....1....0....0....0....0....2....1....1....1....2....1....1....2....0....2..
..0....0....0....1....1....1....1....0....1....2....2....1....2....2....0....1..
..0....2....0....1....1....2....1....2....1....0....0....1....0....2....0....1..

Empirical: T(n,k)=1 if n<=k else sum{i=0..n-k}(binomial(n-k,i)*T(i,k))

Column 1 is A000110
Column 2 is A007476

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

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