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%I #13 Apr 25 2023 01:01:33
%S 1,2,1,5,3,1,15,9,4,1,50,29,14,5,1,176,99,49,20,6,1,638,351,175,76,27,
%T 7,1,2354,1275,637,286,111,35,8,1,8789,4707,2353,1078,441,155,44,9,1,
%U 33099,17577,8788,4081,1728,650,209,54,10,1
%N Triangle read by rows where T(n,k) is the number of nonempty subsets of {1,...,2n-1} with median n and minimum k.
%C The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).
%H Andrew Howroyd, <a href="/A361654/b361654.txt">Table of n, a(n) for n = 1..1275</a> (rows 1..50)
%F T(n,k) = 1 + Sum_{j=1..n-k} binomial(2*j+k-2, j). - _Andrew Howroyd_, Apr 09 2023
%e Triangle begins:
%e 1
%e 2 1
%e 5 3 1
%e 15 9 4 1
%e 50 29 14 5 1
%e 176 99 49 20 6 1
%e 638 351 175 76 27 7 1
%e 2354 1275 637 286 111 35 8 1
%e 8789 4707 2353 1078 441 155 44 9 1
%e Row n = 4 counts the following subsets:
%e {1,7} {2,6} {3,5} {4}
%e {1,4,5} {2,4,5} {3,4,5}
%e {1,4,6} {2,4,6} {3,4,6}
%e {1,4,7} {2,4,7} {3,4,7}
%e {1,2,6,7} {2,3,5,6}
%e {1,3,5,6} {2,3,5,7}
%e {1,3,5,7} {2,3,4,5,6}
%e {1,2,4,5,6} {2,3,4,5,7}
%e {1,2,4,5,7} {2,3,4,6,7}
%e {1,2,4,6,7}
%e {1,3,4,5,6}
%e {1,3,4,5,7}
%e {1,3,4,6,7}
%e {1,2,3,5,6,7}
%e {1,2,3,4,5,6,7}
%t Table[Length[Select[Subsets[Range[2n-1]],Min@@#==k&&Median[#]==n&]],{n,6},{k,n}]
%o (PARI) T(n,k) = sum(j=0, n-k, binomial(2*j+k-2, j)) \\ _Andrew Howroyd_, Apr 09 2023
%Y Row sums appear to be A006134.
%Y Column k = 1 appears to be A024718.
%Y Column k = 2 appears to be A006134.
%Y Column k = 3 appears to be A079309.
%Y A000975 counts subsets with integer median, mean A327475.
%Y A007318 counts subsets by length.
%Y A231147 counts subsets by median, full steps A013580, by mean A327481.
%Y A359893 and A359901 count partitions by median.
%Y A360005(n)/2 gives the median statistic.
%Y Cf. A006134, A057552, A067659, A325347, A359907, A361849.
%K nonn,tabl
%O 1,2
%A _Gus Wiseman_, Mar 23 2023
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