A203435 - OEIS

%I #25 Dec 05 2020 14:20:19

%S 1,1,4,32,392,6883,171088,5661874,242038179,13147317481

%N Number of partitions of {1,2,...,4n} into n 4-element subsets having the same sum.

%C The element sum of each subset is 8n+2.  The larger terms were computed with Knuth's dancing links algorithm.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Dancing_Links">Dancing Links</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>

%e a(1) = 1: {1,2,3,4}.

%e a(2) = 4: {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}.

%p b:= proc() option remember; local i, j, t, m; m:= args[nargs]; if args[1]=0 then `if`(nargs=2, 1, b(args[t] $t=2..nargs)) elif args[1]<1 then 0 else add(`if`(args[j]<m, 0, b(sort([seq(args[i] -`if`(i=j, m+1/97, 0), i=1..nargs-1)])[], m-1)), j=1..nargs-1) fi end:

%p a:= n-> `if`(n=0, 1, b(((8*n+2)+4/97) $n, 4*n)/n!): seq(a(n), n=0..6);

%t b[l_] := 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_] := If[n == 0, 1, b[Join[Array[8*n + 2 + 4/97& , n], {4*n, 4}]]/n!];

%t Table[a[n], {n, 0, 6}] (* _Jean-François Alcover_, Jun 03 2018, adapted from Maple *)

%Y Column k=4 of A203986.

%Y Cf. A104185, A108235, A203017, A264813.

%K nonn,more

%O 0,3

%A _Alois P. Heinz_, Jan 01 2012