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A323529
Number of strict square plane partitions of n.
5
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 5, 7, 11, 13, 19, 23, 31, 37, 47, 55, 69, 79, 95, 109, 129, 145, 169, 189, 217, 241, 273, 301, 339, 371, 413, 451, 499, 541, 595, 643, 703, 757, 823, 925, 999, 1107, 1229, 1387, 1559, 1807, 2071, 2453, 2893, 3451, 4109, 5011
OFFSET
0,11
LINKS
FORMULA
a(n) = Sum_{j>=0} A039622(j) * A008289(n,j^2). - Alois P. Heinz, Jan 24 2019
EXAMPLE
The a(12) = 5 strict square plane partitions:
[12]
.
[1 2] [1 2] [1 3] [1 4]
[3 6] [4 5] [2 6] [2 5]
The a(15) = 13 strict square plane partitions:
[15]
.
[7 5] [8 4] [9 3] [6 5] [7 4] [9 2] [6 4] [7 3] [8 2] [6 3] [6 3] [7 2]
[2 1] [2 1] [2 1] [3 1] [3 1] [3 1] [3 2] [4 1] [4 1] [4 2] [5 1] [5 1]
MAPLE
h:= proc(n) h(n):= (n^2)!*mul(k!/(n+k)!, k=0..n-1) end:
b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2, 0,
`if`(n=0, `if`(issqr(t), h(isqrt(t)), 0),
b(n, i-1, t) +b(n-i, min(n-i, i-1), t+1)))
end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..70); # Alois P. Heinz, Jan 24 2019
MATHEMATICA
Table[Sum[Length[Select[Union[Sort/@Tuples[Reverse/@IntegerPartitions[#, {Length[ptn]}]&/@ptn]], UnsameQ@@Join@@#&&And@@OrderedQ/@Transpose[#]&]], {ptn, IntegerPartitions[n]}], {n, 30}]
(* Second program: *)
h[n_] := (n^2)! Product[k!/(k+n)!, {k, 0, n-1}];
b[n_, i_, t_] := b[n, i, t] = If[n > i(i+1)/2, 0, If[n == 0, If[IntegerQ[ Sqrt[t]], h[Sqrt[t]], 0], b[n-i, Min[n-i, i-1], t+1] + b[n, i-1, t]]];
a[n_] := b[n, n, 0];
a /@ Range[0, 70] (* Jean-François Alcover, May 19 2021, after Alois P. Heinz *)
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 17 2019
EXTENSIONS
More terms from Alois P. Heinz, Jan 24 2019
STATUS
approved