A224524 Table read by antidiagonals: T(n,k) is the number of idempotent n X n 0..k matrices of rank 1.

1, 1, 6, 1, 10, 27, 1, 14, 69, 108, 1, 18, 123, 404, 405, 1, 22, 195, 892, 2155, 1458, 1, 26, 273, 1716, 5845, 10830, 5103, 1, 30, 375, 2732, 13525, 36042, 52241, 17496, 1, 34, 477, 4324, 24575, 99774, 213647, 244648, 59049, 1, 38, 603, 6060, 44545, 208146, 705215, 1232504, 1120599, 196830

Offset: 1

Author

R. H. Hardin, Apr 09 2013

Keywords

• nonn
• tabl

Comments

Table starts

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...

6, 10, 14, 18, 22, 26, 30, 34, 38, ...

27, 69, 123, 195, 273, 375, 477, 603, ...

108, 404, 892, 1716, 2732, 4324, 6060, ...

405, 2155, 5845, 13525, 24575, 44545, ...

1458, 10830, 36042, 99774, 208146, ...

5103, 52241, 213647, 705215, ...

17496, 244648, ...

59049, ...

...

Examples

			Some solutions for n=3, k=4:
  1 0 0   0 4 4   0 0 0   0 4 2   1 2 1   0 0 0   0 1 0
  0 0 0   0 1 1   3 1 0   0 0 0   0 0 0   0 0 0   0 1 0
  1 0 0   0 0 0   0 0 0   0 2 1   0 0 0   1 4 1   0 0 0
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10011

Crossrefs

Column 1 is A027471(n+1).

Programs

Maple

f:= proc(n,k)
  local tot, a1, a0, a2, m,u;
  tot:= 0;
  for a1 from 1 to n do
    for a0 from 0 to n-a1 do
      a2:= n-a1-a0;
      if a0 = 0 then tot:= tot + n!/(a1!*a2!)*a1*(k-1)^a2
      elif a2 = 0 then tot:= tot + n!/(a0!*a1!)*a1*(k+1)^a0
      else
        u:= n!/(a0!*a1!*a2!)*a1;
        for m from 2 to k do
          tot:= tot + u*((m-1)^a2 - (m-2)^a2)*(floor(k/m)+1)^a0
        od
      fi
  od od;
  tot
end proc:
seq(seq(f(i,j-i),i=1..j-1),j=2..20); # Robert Israel, Dec 15 2019

Mathematica

Unprotect[Power]; 0^0 = 1; Protect[Power];
f[n_, k_] := Module[{tot, a1, a0, a2, m, u}, tot = 0; For[a1 = 1, a1 <= n, a1++, For[a0 = 0, a0 <= n - a1, a0++, a2 = n - a1 - a0; Which[a0 == 0, tot = tot + n!/(a1!*a2!)*a1*(k - 1)^a2, a2 == 0, tot = tot + n!/(a0!*a1!)*a1*(k + 1)^a0, True, u = n!/(a0!*a1!*a2!)*a1; For[m = 2, m <= k, m++, tot = tot + u*((m - 1)^a2 - (m - 2)^a2)*(Floor[k/m] + 1)^a0]]]]; tot];
Table[Table[f[i, j - i], {i, 1, j - 1}], {j, 2, 20}] // Flatten (* Jean-François Alcover, Feb 04 2023, after Robert Israel *)