OFFSET
1,1
COMMENTS
Table starts
2 3 4 5 6 7 8 9 10
2 6 12 20 30 42 56 72 90
0 6 24 60 120 210 336 504 720
0 12 72 240 600 1260 2352 4032 6480
0 0 120 720 2520 6720 15120 30240 55440
0 18 408 2940 12600 40110 105168 240408 496080
0 0 840 10080 57960 228480 710640 1874880 4379760
0 24 2448 38640 280560 1338120 4883424 14783328 38962080
0 0 5760 140400 1330560 7761600 33384960 116212320 345945600
0 0 15960 529440 6394680 45291120 228945360 915183360 3075040080
0 66 39864 1956900 30548760 263674950 1568401296 7203324744
0 72 108024 7335840 146516040 1537291560 10751253072
Empirical: row n is a polynomial of degree n.
Coefficients for rows 1-10, highest power first:
1 1
1 1 0
1 0 -1 0
1 0 -1 0 0
1 0 -5 0 4 0
1 0 -6 5 5 -5 0
1 0 -7 0 14 0 -8 0
1 0 -8 0 27 -12 -20 12 0
1 0 -9 0 27 0 -31 0 12 0
1 0 -10 0 35 9 -60 -25 34 16 0
Row n is divisible by n.
Column k is divisible by k+1.
From Robert Israel, Nov 23 2017: (Start)
Row n is a monic polynomial of degree n.
Proof: Let b(j,n,k) be the number of such arrays taking exactly j different values.
Then T(n,k) = Sum_{j <= n} b(j,n,k). But since the j values may be any combination of 0..k taken j at a time, b(j,n,k) = binomial(k+1,j)* b(j,n,j-1) which (if nonzero) is a polynomial in k of degree j.
In particular, b(n,n,n-1) = n!, so b(n,n,k) has degree n and leading coefficient 1. (End)
LINKS
R. H. Hardin, Table of n, a(n) for n = 1..165
EXAMPLE
Some solutions for n=5, k=4:
3 0 1 1 1 0 4 4 0 1 3 2 2 3 1 0
2 4 0 3 0 4 3 2 2 2 4 0 4 4 4 1
0 2 2 2 2 3 0 3 1 4 0 4 3 1 0 0
3 0 3 0 3 1 3 4 4 0 3 0 0 3 4 2
1 3 2 4 0 2 1 0 1 4 2 1 4 0 2 3
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
R. H. Hardin, Aug 06 2012
STATUS
approved