A168238 Number of different 0-moment rowing configurations for 4n rowers.

1, 4, 29, 263, 2724, 30554, 361677, 4454273, 56546511, 735298671, 9749613914, 131377492010, 1794546880363, 24798396567242, 346130144084641, 4873560434459530, 69149450121948083, 987844051312409668, 14198028410251734447, 205181815270346718199

Offset: 1

Author

Jeffrey Shallit, Nov 21 2009

Keywords

• nonn

Comments

Also the number of ways to assign 2n +1's and 2n -1's to the numbers 1,2,...,4n such that the sum is 0, assuming 1 gets the sign +1.

Examples

			For n = 2 there are 4 solutions to the 8-man rowing problem.
For n=1 the unique solution is 1+4 = 2+3. For n=2 there are 4 different solutions: 1+2+7+8 = 3+4+5+6, 1+3+6+8 = 2+4+5+7, 1+4+5+8 = 2+3+6+7, 1+4+6+7 = 2+3+5+8. - _Michel Marcus_, May 25 2013
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..50
• John D. Barrow, Rowing and the Same-Sum Problem Have Their Moments, arXiv:0911.3551 [physics.pop-ph], 2009-2010.

Crossrefs

Bisection of row n=2 of A203986. - Alois P. Heinz, Jan 09 2012

Cf. A227850.

Programs

Mathematica

b[L_List] := b[L] = Module[{nL = Length[L], k = L[[-1]], m = L[[-2]]}, Which[L[[1]] == 0, If[nL == 3, 1, b[L[[2 ;; nL]]]], L[[1]] < 1, 0, True, Sum[If[L[[j]] < m, 0, b[Join[Sort[Table[L[[i]] - If[i == j, m + 1/97, 0], {i, 1, nL - 2}]], {m - 1, k}]]], {j, 1, nL - 2}]]];
A[n_, k_] := If[n==0 || k==0, 1, b[Join[Array[(k (n k + 1)/2 + k/97)&, n], {k n, k}]]/n!];
a[n_] := A[2, 2n];
Array[a, 20] (* Jean-François Alcover, Aug 19 2018, after Alois P. Heinz *)

Extensions

a(6)-a(20) from Alois P. Heinz, Jan 09 2012