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

1,3

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

Links

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

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

Example

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

Crossrefs

Column 1 is A066524

Row 2 is A016825

Sequence in context: A336475 A082744 A127142 * A259671 A224524 A348982

Adjacent sequences: A224330 A224331 A224332 * A224334 A224335 A224336

Keyword

nonn,tabl

Author

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

Status

approved