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Irregular triangle read by rows: T(n,k) is the number of descending plane partitions of order n with the sum of the parts equal to k.
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%I #17 Sep 22 2023 05:36:13

%S 1,1,0,1,1,0,1,1,1,1,1,0,1,1,0,1,1,2,2,3,2,4,3,4,3,4,2,3,2,2,1,1,0,1,

%T 1,0,1,1,2,3,4,4,7,7,10,11,14,14,18,18,21,21,23,22,25,22,23,21,21,18,

%U 18,14,14,11,10,7,7,4,4,3,2,1,1,0,1

%N Irregular triangle read by rows: T(n,k) is the number of descending plane partitions of order n with the sum of the parts equal to k.

%F T(0, 0) = 1, T(1, 0) = 1, T(n, k) = 0 for k < 0 or k > (1/3)*(n+1)*n*(n-1).

%e Rows 1 through 5 are

%e 1;

%e 1, 0, 1;

%e 1, 0, 1, 1, 1, 1, 1, 0, 1;

%e 1, 0, 1, 1, 2, 2, 3, 2, 4, 3, 4, 3, 4, 2, 3, 2, 2, 1, 1, 0, 1;

%e 1, 0, 1, 1, 2, 3, 4, 4, 7, 7, 10, 11, 14, 14, 18, 18, 21, 21, 23, 22, 25, 22, 23, 21, 21, 18, 18, 14, 14, 11, 10, 7, 7, 4, 4, 3, 2, 1, 1, 0, 1.

%Y Row sums give A005130.

%Y Cf. A198890, A005130, A064999.

%K nonn,tabf

%O 1,18

%A _Ludovic Schwob_, Jun 15 2023