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A112972
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Number of ways the set {1,2,...,n} can be split into three subsets of equal sums.
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9
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0, 0, 0, 0, 1, 1, 0, 3, 9, 0, 43, 102, 0, 595, 1480, 0, 9294, 23728, 0, 157991, 411474, 0, 2849968, 7562583, 0, 53987864, 145173095, 0, 1061533318, 2885383960, 0, 21515805520, 59003023409, 0, 447142442841, 1235311936936, 0, 9489835046489, 26382363207307
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OFFSET
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1,8
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LINKS
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FORMULA
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a(n) is 1/6 of the coefficient of x^0*y^0 in Product_{k=1..n} (x^(-2*k)+x^k*(y^k+y^(-k))).
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EXAMPLE
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For n=8 we have 84/75/6321, 84/732/651 and 831/75/642 so a(8)=3.
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MAPLE
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A112972:= n-> coeff(coeff(mul((x^(-2*k)+x^k*(y^k+y^(-k)))
, k=1..n), x, 0), y, 0)/6:
# second Maple program:
b:= proc() option remember; local i, j, t; `if`(args[1]=0,
`if`(nargs=2, 1, b(args[t] $t=2..nargs)), add(
`if`(args[j] -args[nargs]<0, 0, b(sort([seq(args[i]-
`if`(i=j, args[nargs], 0), i=1..nargs-1)])[],
args[nargs]-1)), j=1..nargs-1))
end:
a:= n-> (m-> `if`(irem(m, 3)=0, b((m/3)$3, n)/6, 0))(n*(n+1)/2):
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MATHEMATICA
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b[args_List] := b[args] = Module[{nargs = Length[args]}, If[args[[1]] == 0, If[nargs == 2, 1, b[args // Rest]], Sum[If[args[[j]] - Last[args] < 0, 0, b[Append[Sort[Flatten[Table[args[[i]] - If[i == j, Last[args], 0], {i, 1, nargs-1}]]], Last[args]-1]]], {j, 1, nargs-1}]]];
a[n_] := If[Mod[#, 3] == 0, b[{#/3, #/3, #/3, n}]/6, 0]&[n(n+1)/2];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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