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A133406 Half the number of ways of placing up to n pawns on a length n chessboard row so that the row balances at its middle.

%I A133406 #15 Apr 24 2023 17:00:44
%S A133406 1,1,2,2,4,4,10,9,26,24,76,69,236,214,760,696,2522,2326,8556,7942,
%T A133406 29504,27562,103130,96862,364548,344004,1300820,1232567,4679472,
%U A133406 4449850,16952162,16171118,61790442,59107890,226451036,217157069,833918840
%N A133406 Half the number of ways of placing up to n pawns on a length n chessboard row so that the row balances at its middle.
%C A133406 Odd-indexed terms are A047653.
%C A133406 Also the number of subsets of {1..n-1} that are empty or have mean (n-1)/2. - _Gus Wiseman_, Apr 23 2023
%H A133406 Andrew Howroyd, <a href="/A133406/b133406.txt">Table of n, a(n) for n = 1..500</a>
%F A133406 From _Gus Wiseman_, Apr 23 2023: (Start)
%F A133406 a(2n+1) = A000980(n)/2 = A047653(n).
%F A133406 a(n) = A362046(n-1) + 1.
%F A133406 (End)
%e A133406 From _Gus Wiseman_, Apr 23 2023: (Start)
%e A133406 The a(1) = 1 through a(8) = 9 subsets:
%e A133406   {}  {}  {}   {}     {}       {}         {}           {}
%e A133406           {1}  {1,2}  {2}      {1,4}      {3}          {1,6}
%e A133406                       {1,3}    {2,3}      {1,5}        {2,5}
%e A133406                       {1,2,3}  {1,2,3,4}  {2,4}        {3,4}
%e A133406                                           {1,2,6}      {1,2,4,7}
%e A133406                                           {1,3,5}      {1,2,5,6}
%e A133406                                           {2,3,4}      {1,3,4,6}
%e A133406                                           {1,2,3,6}    {2,3,4,5}
%e A133406                                           {1,2,4,5}    {1,2,3,4,5,6}
%e A133406                                           {1,2,3,4,5}
%e A133406 (End)
%t A133406 Table[Length[Select[Subsets[Range[n]],Length[#]==0||Mean[#]==n/2&]],{n,0,10}] (* _Gus Wiseman_, Apr 23 2023 *)
%o A133406 (PARI) a(n) = {polcoef(prod(k=1, n, 1 + 'x^(2*k-n-1)), 0)/2} \\ _Andrew Howroyd_, Jan 07 2023
%Y A133406 Cf. A047653, A133413, A133414.
%Y A133406 For median instead of mean we have A361801 + 1, the doubling of A024718.
%Y A133406 Not counting the empty set gives A362046 (shifted left).
%Y A133406 A007318 counts subsets by length, A327481 by integer mean.
%Y A133406 A047653 counts subsets of {1..2n} with mean n, nonempty A212352.
%Y A133406 A070925 counts subsets of {1..2n-1} with mean n, nonempty A000980.
%Y A133406 A327475 counts subsets with integer mean, nonempty A051293.
%Y A133406 Cf. A000975, A002838, A047997, A057552, A079309.
%K A133406 nonn
%O A133406 1,3
%A A133406 _R. H. Hardin_, Nov 24 2007