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A361654 Triangle read by rows where T(n,k) is the number of nonempty subsets of {1,...,2n-1} with median n and minimum k.

%I A361654 #21 Feb 19 2025 12:11:52
%S A361654 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 A361654 7,1,2354,1275,637,286,111,35,8,1,8789,4707,2353,1078,441,155,44,9,1,
%U A361654 33099,17577,8788,4081,1728,650,209,54,10,1
%N A361654 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 A361654 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 A361654 Andrew Howroyd, <a href="/A361654/b361654.txt">Table of n, a(n) for n = 1..1275</a> (rows 1..50)
%H A361654 Paul Barry, <a href="https://arxiv.org/abs/2409.09547">A Riordan array family for some integrable lattice models</a>, arXiv:2409.09547 [math.CO], 2024. See p. 7.
%H A361654 Paul Barry, <a href="https://doi.org/10.3390/math13020242">Extensions of Riordan Arrays and Their Applications</a>, Mathematics (2025) Vol. 13, No. 2, 242. See p. 12.
%F A361654 T(n,k) = 1 + Sum_{j=1..n-k} binomial(2*j+k-2, j). - _Andrew Howroyd_, Apr 09 2023
%e A361654 Triangle begins:
%e A361654      1
%e A361654      2     1
%e A361654      5     3     1
%e A361654     15     9     4     1
%e A361654     50    29    14     5     1
%e A361654    176    99    49    20     6     1
%e A361654    638   351   175    76    27     7     1
%e A361654   2354  1275   637   286   111    35     8     1
%e A361654   8789  4707  2353  1078   441   155    44     9     1
%e A361654 Row n = 4 counts the following subsets:
%e A361654   {1,7}            {2,6}        {3,5}    {4}
%e A361654   {1,4,5}          {2,4,5}      {3,4,5}
%e A361654   {1,4,6}          {2,4,6}      {3,4,6}
%e A361654   {1,4,7}          {2,4,7}      {3,4,7}
%e A361654   {1,2,6,7}        {2,3,5,6}
%e A361654   {1,3,5,6}        {2,3,5,7}
%e A361654   {1,3,5,7}        {2,3,4,5,6}
%e A361654   {1,2,4,5,6}      {2,3,4,5,7}
%e A361654   {1,2,4,5,7}      {2,3,4,6,7}
%e A361654   {1,2,4,6,7}
%e A361654   {1,3,4,5,6}
%e A361654   {1,3,4,5,7}
%e A361654   {1,3,4,6,7}
%e A361654   {1,2,3,5,6,7}
%e A361654   {1,2,3,4,5,6,7}
%t A361654 Table[Length[Select[Subsets[Range[2n-1]],Min@@#==k&&Median[#]==n&]],{n,6},{k,n}]
%o A361654 (PARI) T(n,k) = sum(j=0, n-k, binomial(2*j+k-2, j)) \\ _Andrew Howroyd_, Apr 09 2023
%Y A361654 Row sums appear to be A006134.
%Y A361654 Column k = 1 appears to be A024718.
%Y A361654 Column k = 2 appears to be A006134.
%Y A361654 Column k = 3 appears to be A079309.
%Y A361654 A000975 counts subsets with integer median, mean A327475.
%Y A361654 A007318 counts subsets by length.
%Y A361654 A231147 counts subsets by median, full steps A013580, by mean A327481.
%Y A361654 A359893 and A359901 count partitions by median.
%Y A361654 A360005(n)/2 gives the median statistic.
%Y A361654 Cf. A006134, A057552, A067659, A325347, A359907, A361849.
%K A361654 nonn,tabl
%O A361654 1,2
%A A361654 _Gus Wiseman_, Mar 23 2023