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A323522
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Number of ways to fill a square matrix with the parts of a strict integer partition of n.
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4
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1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 25, 25, 49, 73, 121, 145, 217, 265, 361, 433, 553, 649, 817, 937, 1129, 1297, 1537, 1729, 2017, 2257, 2593, 2881, 3265, 3601, 4057, 4441, 4945, 5401, 5977, 6481, 7129, 7705, 8425, 9073, 9865, 373465, 374353, 738025, 1101865, 1828513
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OFFSET
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0,11
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LINKS
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FORMULA
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a(n) = Sum_{k >= 0} (k^2)! * Q(n, k^2) where Q = A008289.
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EXAMPLE
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The a(10) = 25 matrices:
[10]
.
[4 3] [4 3] [4 2] [4 2] [4 1] [4 1] [3 4] [3 4]
[2 1] [1 2] [3 1] [1 3] [3 2] [2 3] [2 1] [1 2]
.
[3 2] [3 2] [3 1] [3 1] [2 4] [2 4] [2 3] [2 3]
[4 1] [1 4] [4 2] [2 4] [3 1] [1 3] [4 1] [1 4]
.
[2 1] [2 1] [1 4] [1 4] [1 3] [1 3] [1 2] [1 2]
[4 3] [3 4] [3 2] [2 3] [4 2] [2 4] [4 3] [3 4]
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MAPLE
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b:= proc(n, i) b(n, i):= `if`(n=0, [1], `if`(i<1, [], zip((x, y)
-> x+y, b(n, i-1), `if`(i>n, [], [0, b(n-i, i-1)[]]), 0)))
end:
a:= n-> (l-> add(l[i^2+1]*(i^2)!, i=0..floor(sqrt(nops(l)-1))))(b(n$2)):
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MATHEMATICA
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Table[Sum[(k^2)!*Length[Select[IntegerPartitions[n, {k^2}], UnsameQ@@#&]], {k, n}], {n, 20}]
(* Second program: *)
q[n_, k_] := q[n, k] = If[n < k || k < 1, 0,
If[n == 1, 1, q[n-k, k] + q[n-k, k-1]]];
a[n_] := If[n == 0, 1, Sum[(k^2)! q[n, k^2], {k, 0, n}]];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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