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A224524
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Table read by antidiagonals: T(n,k) is the number of idempotent n X n 0..k matrices of rank 1.
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5
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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
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OFFSET
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1,3
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COMMENTS
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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, ...
...
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LINKS
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EXAMPLE
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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
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MAPLE
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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
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MATHEMATICA
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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];
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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