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Irregular triangle read by rows where T(n,k) is the number of k-subsets of {-n+1,...,n} with sum 0, for k = 1,...,2n-1.
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%I #11 Apr 11 2023 22:40:54

%S 1,1,1,1,1,2,3,2,1,1,3,6,7,5,2,1,1,4,10,16,18,14,8,3,1,1,5,15,31,46,

%T 51,43,27,12,3,1,1,6,21,53,98,139,155,134,88,43,16,4,1,1,7,28,83,184,

%U 319,441,486,424,293,161,68,21,4,1

%N Irregular triangle read by rows where T(n,k) is the number of k-subsets of {-n+1,...,n} with sum 0, for k = 1,...,2n-1.

%C Also the number of k-subsets of {1,...,2n} with mean n.

%e Triangle begins:

%e 1

%e 1 1 1

%e 1 2 3 2 1

%e 1 3 6 7 5 2 1

%e 1 4 10 16 18 14 8 3 1

%e 1 5 15 31 46 51 43 27 12 3 1

%e 1 6 21 53 98 139 155 134 88 43 16 4 1

%e 1 7 28 83 184 319 441 486 424 293 161 68 21 4 1

%e Row n = 3 counts the following subsets:

%e {0} {-1,1} {-1,0,1} {-2,-1,0,3} {-2,-1,0,1,2}

%e {-2,2} {-2,0,2} {-2,-1,1,2}

%e {-2,-1,3}

%t Table[Length[Select[Subsets[Range[-n+1,n],{k}],Total[#]==0&]],{n,6},{k,2n-1}]

%Y Row lengths are A005408.

%Y Row sums are A212352.

%Y A007318 counts subsets by length.

%Y A067538 counts partitions with integer mean.

%Y A231147 counts subsets by median.

%Y A327475 counts subsets with integer mean, median A000975.

%Y A327481 counts subsets by mean.

%Y Cf. A006134, A013580, A024718, A079309, A326512, A349156, A361654, A362046.

%K nonn,tabf

%O 1,6

%A _Gus Wiseman_, Apr 10 2023