A215228 T(n,k) = number of length-n 0..k arrays connected end-around, with no sequence of L

2, 3, 2, 4, 6, 0, 5, 12, 6, 0, 6, 20, 24, 12, 0, 7, 30, 60, 72, 0, 0, 8, 42, 120, 240, 120, 18, 0, 9, 56, 210, 600, 720, 408, 0, 0, 10, 72, 336, 1260, 2520, 2940, 840, 24, 0, 11, 90, 504, 2352, 6720, 12600, 10080, 2448, 0, 0, 12, 110, 720, 4032, 15120, 40110, 57960, 38640

Offset: 1

Author

R. H. Hardin, Aug 06 2012

Keywords

• nonn
• tabl

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)

Examples

			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
		

Links

• R. H. Hardin, Table of n, a(n) for n = 1..165

Crossrefs

Column 2 is A066297.

Row 2 is A002378.

Row 3 is A007531(n+1).

Row 4 is A047928(n+1).

Row 5 is A052787(n+2).