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%I #19 Jan 18 2019 14:20:50
%S 1,1,1,1,2,2,4,5,8,11,16,21,31,41,57,78,108,146,202,274,375,509,690,
%T 929,1255,1679,2246,2991,3979,5266,6971,9187,12104,15898,20870,27322,
%U 35762,46690,60927,79348,103270,134138,174108,225576,291990,377320,487083
%N Number of square plane partitions of n.
%C Number of ways of writing n as a sum p(1,1) + p(1,2) + ... + p(1,k) + p(2,1) + ... + p(2,k) + ... + p(k,1) + ... + p(k,k) for some k so that in the square array {p(i,j)} the numbers are nonincreasing along rows and columns. All the p(i,j) are >= 1.
%H Alois P. Heinz, <a href="/A089299/b089299.txt">Table of n, a(n) for n = 0..4000</a>
%F G.f.: Sum_{k>=0} x^(k^2) / Product_{j=1..2k-1} (1-x^j)^min(j,2k-j). - _Franklin T. Adams-Watters_, Jun 14 2006
%e a(7) = 5:
%e 7 41 32 31 22
%e . 11 11 21 21
%e a(10) = 16 from {{10}}, {{3, 2}, {3, 2}}, {{3, 3}, {2, 2}}, {{3, 3}, {3, 1}}, {{4, 1}, {4, 1}}, {{4, 2}, {2, 2}}, {{4, 2}, {3, 1}}, {{4, 3}, {2, 1}}, {{4, 4}, {1, 1}}, {{5, 1}, {3, 1}}, {{5, 2}, {2, 1}}, {{5, 3}, {1, 1}}, {{6, 1}, {2, 1}}, {{6, 2}, {1, 1}}, {{7, 1}, {1, 1}}, {{2, 1, 1}, {1, 1, 1}, {1, 1, 1}}}
%e From _Gus Wiseman_, Jan 16 2019: (Start)
%e The a(10) = 16 square plane partitions:
%e [ten]
%e .
%e [32] [33] [33] [41] [42] [42] [43] [44] [51] [52] [53] [61] [62] [71]
%e [32] [22] [31] [41] [22] [31] [21] [11] [31] [21] [11] [21] [11] [11]
%e .
%e [211]
%e [111]
%e [111]
%e (End)
%t Table[Sum[Length[Select[Union[Sort/@Tuples[IntegerPartitions[#,{Length[ptn]}]&/@ptn]],And@@OrderedQ/@Transpose[#]&]],{ptn,IntegerPartitions[n]}],{n,30}] (* _Gus Wiseman_, Jan 16 2019 *)
%Y Cf. A008763, A001970, A089292.
%Y Cf. A000219, A003293, A101509, A319066, A323429, A323433, A323450.
%K nonn
%O 0,5
%A _N. J. A. Sloane_, Dec 25 2003
%E Corrected and extended by _Wouter Meeussen_, Dec 30 2003
%E a(21)-a(25) from _John W. Layman_, Jan 02 2004
%E More terms from _Franklin T. Adams-Watters_, Jun 14 2006
%E Name edited by _Gus Wiseman_, Jan 16 2019
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