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%I #31 Aug 19 2018 10:42:58
%S 1,4,29,263,2724,30554,361677,4454273,56546511,735298671,9749613914,
%T 131377492010,1794546880363,24798396567242,346130144084641,
%U 4873560434459530,69149450121948083,987844051312409668,14198028410251734447,205181815270346718199
%N Number of different 0-moment rowing configurations for 4n rowers.
%C 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.
%H Alois P. Heinz, <a href="/A168238/b168238.txt">Table of n, a(n) for n = 1..50</a>
%H John D. Barrow, <a href="http://arxiv.org/abs/0911.3551">Rowing and the Same-Sum Problem Have Their Moments</a>, arXiv:0911.3551 [physics.pop-ph], 2009-2010.
%e For n = 2 there are 4 solutions to the 8-man rowing problem.
%e 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
%t 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}]]];
%t A[n_, k_] := If[n==0 || k==0, 1, b[Join[Array[(k (n k + 1)/2 + k/97)&, n], {k n, k}]]/n!];
%t a[n_] := A[2, 2n];
%t Array[a, 20] (* _Jean-François Alcover_, Aug 19 2018, after _Alois P. Heinz_ *)
%Y Bisection of row n=2 of A203986. - _Alois P. Heinz_, Jan 09 2012
%Y Cf. A227850.
%K nonn
%O 1,2
%A _Jeffrey Shallit_, Nov 21 2009
%E a(6)-a(20) from _Alois P. Heinz_, Jan 09 2012
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