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A224524 Table read by antidiagonals: T(n,k) is the number of idempotent n X n 0..k matrices of rank 1. 5
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,3
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, ...
...
LINKS
EXAMPLE
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
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 *)
CROSSREFS
Column 1 is A027471(n+1).
Sequence in context: A127142 A224333 A259671 * A348982 A350677 A046618
KEYWORD
nonn,tabl
AUTHOR
R. H. Hardin, Apr 09 2013
STATUS
approved

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Last modified April 13 20:16 EDT 2024. Contains 371645 sequences. (Running on oeis4.)