A224333 T(n,k)=Number of idempotent n X n 0..k matrices of rank n-1.

1, 1, 6, 1, 10, 21, 1, 14, 51, 60, 1, 18, 93, 212, 155, 1, 22, 147, 508, 805, 378, 1, 26, 213, 996, 2555, 2910, 889, 1, 30, 291, 1724, 6245, 12282, 10199, 2040, 1, 34, 381, 2740, 12955, 37494, 57337, 34984, 4599, 1, 38, 483, 4092, 24005, 93306, 218743, 262136

Offset: 1

Author

R. H. Hardin, formula via M. F. Hasler _William J. Keith_ and Rob Pratt in the Sequence Fans Mailing List, Apr 03 2013

Keywords

• nonn
• tabl

Comments

Table starts

.....1......1.......1........1.........1.........1..........1..........1

.....6.....10......14.......18........22........26.........30.........34

....21.....51......93......147.......213.......291........381........483

....60....212.....508......996......1724......2740.......4092.......5828

...155....805....2555.....6245.....12955.....24005......40955......65605

...378...2910...12282....37494.....93306....201678.....393210.....708582

...889..10199...57337...218743....653177...1647079....3670009....7440167

..2040..34984..262136..1249992...4478968..13176680...33554424...76527496

..4599.118089.1179639..7031241..30233079.103766409..301989879..774840969

.10230.393650.5242870.39062490.201553910.807072130.2684354550.7748409770

Examples

			Some solutions for n=3 k=4
..1..1..0....0..0..0....1..0..0....1..0..4....1..0..0....0..3..1....0..3..0
..0..0..0....3..1..0....0..0..2....0..1..1....0..1..0....0..1..0....0..1..0
..0..2..1....1..0..1....0..0..1....0..0..0....1..2..0....0..0..1....0..0..1
		

Links

• R. H. Hardin, Table of n, a(n) for n = 1..1000

Crossrefs

Column 1 is A066524

Row 2 is A016825

Formula

T(n,k) = 2*n*(1+k)^(n-1)-n

For column k:

k=1: a(n) = 6*a(n-1) -13*a(n-2) +12*a(n-3) -4*a(n-4)

k=2: a(n) = 8*a(n-1) -22*a(n-2) +24*a(n-3) -9*a(n-4)

k=3: a(n) = 10*a(n-1) -33*a(n-2) +40*a(n-3) -16*a(n-4)

k=4: a(n) = 12*a(n-1) -46*a(n-2) +60*a(n-3) -25*a(n-4)

k=5: a(n) = 14*a(n-1) -61*a(n-2) +84*a(n-3) -36*a(n-4)

k=6: a(n) = 16*a(n-1) -78*a(n-2) +112*a(n-3) -49*a(n-4)

k=7: a(n) = 18*a(n-1) -97*a(n-2) +144*a(n-3) -64*a(n-4)

For row n:

n=1: a(n) = 1

n=2: a(n) = 4*n + 2

n=3: a(n) = 6*n^2 + 12*n + 3

n=4: a(n) = 8*n^3 + 24*n^2 + 24*n + 4

n=5: a(n) = 10*n^4 + 40*n^3 + 60*n^2 + 40*n + 5

n=6: a(n) = 12*n^5 + 60*n^4 + 120*n^3 + 120*n^2 + 60*n + 6

n=7: a(n) = 14*n^6 + 84*n^5 + 210*n^4 + 280*n^3 + 210*n^2 + 84*n + 7