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A361802 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.

%I A361802 #11 Apr 11 2023 22:40:54
%S A361802 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 A361802 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 A361802 319,441,486,424,293,161,68,21,4,1
%N A361802 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 A361802 Also the number of k-subsets of {1,...,2n} with mean n.
%e A361802 Triangle begins:
%e A361802    1
%e A361802    1   1   1
%e A361802    1   2   3   2   1
%e A361802    1   3   6   7   5   2   1
%e A361802    1   4  10  16  18  14   8   3   1
%e A361802    1   5  15  31  46  51  43  27  12   3   1
%e A361802    1   6  21  53  98 139 155 134  88  43  16   4   1
%e A361802    1   7  28  83 184 319 441 486 424 293 161  68  21   4   1
%e A361802 Row n = 3 counts the following subsets:
%e A361802   {0}  {-1,1}  {-1,0,1}   {-2,-1,0,3}  {-2,-1,0,1,2}
%e A361802        {-2,2}  {-2,0,2}   {-2,-1,1,2}
%e A361802                {-2,-1,3}
%t A361802 Table[Length[Select[Subsets[Range[-n+1,n],{k}],Total[#]==0&]],{n,6},{k,2n-1}]
%Y A361802 Row lengths are A005408.
%Y A361802 Row sums are A212352.
%Y A361802 A007318 counts subsets by length.
%Y A361802 A067538 counts partitions with integer mean.
%Y A361802 A231147 counts subsets by median.
%Y A361802 A327475 counts subsets with integer mean, median A000975.
%Y A361802 A327481 counts subsets by mean.
%Y A361802 Cf. A006134, A013580, A024718, A079309, A326512, A349156, A361654, A362046.
%K A361802 nonn,tabf
%O A361802 1,6
%A A361802 _Gus Wiseman_, Apr 10 2023