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

%I A224333 #7 Jun 02 2025 08:30:59
%S A224333 1,1,6,1,10,21,1,14,51,60,1,18,93,212,155,1,22,147,508,805,378,1,26,
%T A224333 213,996,2555,2910,889,1,30,291,1724,6245,12282,10199,2040,1,34,381,
%U A224333 2740,12955,37494,57337,34984,4599,1,38,483,4092,24005,93306,218743,262136
%N A224333 T(n,k)=Number of idempotent n X n 0..k matrices of rank n-1.
%C A224333 Table starts
%C A224333 .....1......1.......1........1.........1.........1..........1..........1
%C A224333 .....6.....10......14.......18........22........26.........30.........34
%C A224333 ....21.....51......93......147.......213.......291........381........483
%C A224333 ....60....212.....508......996......1724......2740.......4092.......5828
%C A224333 ...155....805....2555.....6245.....12955.....24005......40955......65605
%C A224333 ...378...2910...12282....37494.....93306....201678.....393210.....708582
%C A224333 ...889..10199...57337...218743....653177...1647079....3670009....7440167
%C A224333 ..2040..34984..262136..1249992...4478968..13176680...33554424...76527496
%C A224333 ..4599.118089.1179639..7031241..30233079.103766409..301989879..774840969
%C A224333 .10230.393650.5242870.39062490.201553910.807072130.2684354550.7748409770
%H A224333 R. H. Hardin, <a href="/A224333/b224333.txt">Table of n, a(n) for n = 1..1000</a>
%F A224333 T(n,k) = 2*n*(1+k)^(n-1)-n
%F A224333 For column k:
%F A224333 k=1: a(n) = 6*a(n-1) -13*a(n-2) +12*a(n-3) -4*a(n-4)
%F A224333 k=2: a(n) = 8*a(n-1) -22*a(n-2) +24*a(n-3) -9*a(n-4)
%F A224333 k=3: a(n) = 10*a(n-1) -33*a(n-2) +40*a(n-3) -16*a(n-4)
%F A224333 k=4: a(n) = 12*a(n-1) -46*a(n-2) +60*a(n-3) -25*a(n-4)
%F A224333 k=5: a(n) = 14*a(n-1) -61*a(n-2) +84*a(n-3) -36*a(n-4)
%F A224333 k=6: a(n) = 16*a(n-1) -78*a(n-2) +112*a(n-3) -49*a(n-4)
%F A224333 k=7: a(n) = 18*a(n-1) -97*a(n-2) +144*a(n-3) -64*a(n-4)
%F A224333 For row n:
%F A224333 n=1: a(n) = 1
%F A224333 n=2: a(n) = 4*n + 2
%F A224333 n=3: a(n) = 6*n^2 + 12*n + 3
%F A224333 n=4: a(n) = 8*n^3 + 24*n^2 + 24*n + 4
%F A224333 n=5: a(n) = 10*n^4 + 40*n^3 + 60*n^2 + 40*n + 5
%F A224333 n=6: a(n) = 12*n^5 + 60*n^4 + 120*n^3 + 120*n^2 + 60*n + 6
%F A224333 n=7: a(n) = 14*n^6 + 84*n^5 + 210*n^4 + 280*n^3 + 210*n^2 + 84*n + 7
%e A224333 Some solutions for n=3 k=4
%e A224333 ..1..1..0....0..0..0....1..0..0....1..0..4....1..0..0....0..3..1....0..3..0
%e A224333 ..0..0..0....3..1..0....0..0..2....0..1..1....0..1..0....0..1..0....0..1..0
%e A224333 ..0..2..1....1..0..1....0..0..1....0..0..0....1..2..0....0..0..1....0..0..1
%Y A224333 Column 1 is A066524
%Y A224333 Row 2 is A016825
%K A224333 nonn,tabl
%O A224333 1,3
%A A224333 _R. H. Hardin_, formula via _M. F. Hasler_ _William J. Keith_ and Rob Pratt in the Sequence Fans Mailing List, Apr 03 2013