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.

1, 1, 2, 2, 4, 4, 10, 9, 26, 24, 76, 69, 236, 214, 760, 696, 2522, 2326, 8556, 7942, 29504, 27562, 103130, 96862, 364548, 344004, 1300820, 1232567, 4679472, 4449850, 16952162, 16171118, 61790442, 59107890, 226451036, 217157069, 833918840

Offset: 1

Author

R. H. Hardin, Nov 24 2007

Keywords

• nonn

Comments

Odd-indexed terms are A047653.

Also the number of subsets of {1..n-1} that are empty or have mean (n-1)/2. - Gus Wiseman, Apr 23 2023

Examples

			From _Gus Wiseman_, Apr 23 2023: (Start)
The a(1) = 1 through a(8) = 9 subsets:
  {}  {}  {}   {}     {}       {}         {}           {}
          {1}  {1,2}  {2}      {1,4}      {3}          {1,6}
                      {1,3}    {2,3}      {1,5}        {2,5}
                      {1,2,3}  {1,2,3,4}  {2,4}        {3,4}
                                          {1,2,6}      {1,2,4,7}
                                          {1,3,5}      {1,2,5,6}
                                          {2,3,4}      {1,3,4,6}
                                          {1,2,3,6}    {2,3,4,5}
                                          {1,2,4,5}    {1,2,3,4,5,6}
                                          {1,2,3,4,5}
(End)
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..500

Crossrefs

Cf. A047653, A133413, A133414.

For median instead of mean we have A361801 + 1, the doubling of A024718.

Not counting the empty set gives A362046 (shifted left).

A007318 counts subsets by length, A327481 by integer mean.

A047653 counts subsets of {1..2n} with mean n, nonempty A212352.

A070925 counts subsets of {1..2n-1} with mean n, nonempty A000980.

A327475 counts subsets with integer mean, nonempty A051293.

Cf. A000975, A002838, A047997, A057552, A079309.

Programs

Mathematica

Table[Length[Select[Subsets[Range[n]],Length[#]==0||Mean[#]==n/2&]],{n,0,10}] (* Gus Wiseman, Apr 23 2023 *)

PARI

a(n) = {polcoef(prod(k=1, n, 1 + 'x^(2*k-n-1)), 0)/2} \\ Andrew Howroyd, Jan 07 2023

Formula

From Gus Wiseman, Apr 23 2023: (Start)

a(2n+1) = A000980(n)/2 = A047653(n).

a(n) = A362046(n-1) + 1.

(End)