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A319600 Number T(n,k) of plane partitions of n into parts of exactly k sorts; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

%I A319600 #19 Feb 18 2025 08:10:44
%S A319600 1,0,1,0,3,4,0,6,22,18,0,13,96,198,120,0,24,330,1272,1800,840,0,48,
%T A319600 1146,7518,19152,20640,7920,0,86,3518,36684,148200,274080,234720,
%U A319600 75600,0,160,10946,177438,1080960,3083640,4462560,3180240,887040,0,282,32102,788928,6952440,28621920,62056080,73175760,44432640,10886400
%N A319600 Number T(n,k) of plane partitions of n into parts of exactly k sorts; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%H A319600 Alois P. Heinz, <a href="/A319600/b319600.txt">Rows n = 0..50, flattened</a>
%H A319600 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PlanePartition.html">Plane partition</a>
%H A319600 Wikipedia, <a href="https://en.wikipedia.org/wiki/Plane_partition">Plane partition</a>
%F A319600 T(n,k) = Sum_{i=0..k} (-1)^i * C(k,i) * A306100(n,k-i).
%F A319600 T(n,k) = k! * A319730(n,k).
%e A319600 Triangle T(n,k) begins:
%e A319600   1;
%e A319600   0,   1;
%e A319600   0,   3,     4;
%e A319600   0,   6,    22,     18;
%e A319600   0,  13,    96,    198,     120;
%e A319600   0,  24,   330,   1272,    1800,     840;
%e A319600   0,  48,  1146,   7518,   19152,   20640,    7920;
%e A319600   0,  86,  3518,  36684,  148200,  274080,  234720,   75600;
%e A319600   0, 160, 10946, 177438, 1080960, 3083640, 4462560, 3180240, 887040;
%e A319600   ...
%Y A319600 Columns k=0-1 give: A000007, A000219 (for n>0).
%Y A319600 Row sums give A319601.
%Y A319600 Main diagonal gives A053529.
%Y A319600 Cf. A255970, A319730.
%K A319600 nonn,tabl
%O A319600 0,5
%A A319600 _Alois P. Heinz_, Sep 24 2018