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A323529 Number of strict square plane partitions of n.

%I A323529 #21 May 19 2021 05:14:35
%S A323529 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,
%T A323529 145,169,189,217,241,273,301,339,371,413,451,499,541,595,643,703,757,
%U A323529 823,925,999,1107,1229,1387,1559,1807,2071,2453,2893,3451,4109,5011
%N A323529 Number of strict square plane partitions of n.
%H A323529 Alois P. Heinz, <a href="/A323529/b323529.txt">Table of n, a(n) for n = 0..7500</a>
%F A323529 a(n) = Sum_{j>=0} A039622(j) * A008289(n,j^2). - _Alois P. Heinz_, Jan 24 2019
%e A323529 The a(12) = 5 strict square plane partitions:
%e A323529   [12]
%e A323529 .
%e A323529   [1 2] [1 2] [1 3] [1 4]
%e A323529   [3 6] [4 5] [2 6] [2 5]
%e A323529 The a(15) = 13 strict square plane partitions:
%e A323529   [15]
%e A323529 .
%e A323529   [7 5] [8 4] [9 3] [6 5] [7 4] [9 2] [6 4] [7 3] [8 2] [6 3] [6 3] [7 2]
%e A323529   [2 1] [2 1] [2 1] [3 1] [3 1] [3 1] [3 2] [4 1] [4 1] [4 2] [5 1] [5 1]
%p A323529 h:= proc(n) h(n):= (n^2)!*mul(k!/(n+k)!, k=0..n-1) end:
%p A323529 b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2, 0,
%p A323529       `if`(n=0, `if`(issqr(t), h(isqrt(t)), 0),
%p A323529          b(n, i-1, t) +b(n-i, min(n-i, i-1), t+1)))
%p A323529     end:
%p A323529 a:= n-> b(n$2, 0):
%p A323529 seq(a(n), n=0..70);  # _Alois P. Heinz_, Jan 24 2019
%t A323529 Table[Sum[Length[Select[Union[Sort/@Tuples[Reverse/@IntegerPartitions[#,{Length[ptn]}]&/@ptn]],UnsameQ@@Join@@#&&And@@OrderedQ/@Transpose[#]&]],{ptn,IntegerPartitions[n]}],{n,30}]
%t A323529 (* Second program: *)
%t A323529 h[n_] := (n^2)! Product[k!/(k+n)!, {k, 0, n-1}];
%t A323529 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]]];
%t A323529 a[n_] := b[n, n, 0];
%t A323529 a /@ Range[0, 70] (* _Jean-François Alcover_, May 19 2021, after _Alois P. Heinz_ *)
%Y A323529 Cf. A000219, A008289, A039622, A047966, A089299, A101509, A319066, A323429, A323434, A323522, A323530.
%K A323529 nonn
%O A323529 0,11
%A A323529 _Gus Wiseman_, Jan 17 2019
%E A323529 More terms from _Alois P. Heinz_, Jan 24 2019