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A112972 Number of ways the set {1,2,...,n} can be split into three subsets of equal sums. 9
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,8

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..125

FORMULA

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))).

EXAMPLE

For n=8 we have 84/75/6321, 84/732/651 and 831/75/642 so a(8)=3.

MAPLE

A112972:= n-> coeff(coeff(mul((x^(-2*k)+x^k*(y^k+y^(-k)))

              , k=1..n), x, 0), y, 0)/6:

seq(A112972(n), n=1..20);

# 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):

seq(a(n), n=1..42);  # Alois P. Heinz, Sep 03 2009

MATHEMATICA

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];

Array[a, 42] (* Jean-Fran├žois Alcover, Oct 30 2020, after Alois P. Heinz *)

CROSSREFS

Cf. A112956, A058377.

Column k=3 of A275714.

Similar sequences: A327448, A327449, A327450.

Sequence in context: A021723 A343618 A206160 * A334191 A258147 A016673

Adjacent sequences:  A112969 A112970 A112971 * A112973 A112974 A112975

KEYWORD

nonn

AUTHOR

Floor van Lamoen, Oct 07 2005

EXTENSIONS

Extended beyond a(25) by Alois P. Heinz, Sep 03 2009

STATUS

approved

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Last modified May 17 09:03 EDT 2021. Contains 343969 sequences. (Running on oeis4.)