A302919 The number of ways of placing 2n-1 white balls and 2n-1 black balls into unlabeled bins such that each bin has both an odd number of white balls and black balls.

1, 2, 4, 12, 32, 85, 217, 539, 1316, 3146, 7374, 16969, 38387, 85452, 187456, 405659, 866759, 1830086, 3821072, 7894447, 16148593, 32723147, 65719405, 130871128, 258513076, 506724988, 985968770, 1904992841, 3655873294, 6970687150, 13208622956, 24879427889, 46593011280, 86773920240, 160742462714, 296227087942, 543183754454, 991213989213

Offset

1,2

Links

Table of n, a(n) for n=1..38.

Marko Riedel, How many ways to get an odd number of each color in each bin?, Mathematics Stack Exchange.

Example

For n = 3 the a(3) = 4 ways to place five white and five black balls are (wwwwwbbbbb), (wwwbbb)(wb)(wb), (wwwb)(wbbb)(wb), and (wb)(wb)(wb)(wb)(wb).

Mathematica

nmax = 15; p = 1; Do[Do[p = Expand[p*(1 - x^(2*i - 1)*y^(2*j - 1))]; p = Select[p, (Exponent[#, x] <= 2*nmax - 1) && (Exponent[#, y] <= 2*nmax - 1) &], {i, 1, 2*nmax - 1}], {j, 1, nmax}]; p = Expand[Normal[Series[1/p, {x, 0, 2*nmax - 1}, {y, 0, 2*nmax - 1}]]]; p = Select[p, Exponent[#, x] == Exponent[#, y] &]; Flatten[{1, Table[Coefficient[p, x^(2*n - 1)*y^(2*n - 1)], {n, 1, nmax}]}] (* Vaclav Kotesovec, Apr 16 2018 *)

Crossrefs

Cf. A090806, A108469.

Sequence in context: A323864 A242659 A109388 * A181329 A293007 A028860

Adjacent sequences: A302916 A302917 A302918 * A302920 A302921 A302922

Keyword

nonn

Author

Peter Kagey, Apr 15 2018

Extensions

More terms from Vaclav Kotesovec, Apr 16 2018

Status

approved