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

1,3

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

Links

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

Formula

From Gus Wiseman, Apr 23 2023: (Start)

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

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

(End)

Example

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)

Mathematica

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

Prog

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

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.

Sequence in context: A231523 A114338 A170831 * A240381 A295746 A272104

Adjacent sequences: A133403 A133404 A133405 * A133407 A133408 A133409

Keyword

nonn

Author

R. H. Hardin, Nov 24 2007

Status

approved