A113039 Number of ways the set {1,2,...,n} can be split into three subsets of which the three sums are consecutive.

0, 0, 1, 0, 3, 5, 0, 23, 52, 0, 254, 593, 0, 3611, 8859, 0, 55554, 142169, 0, 946871, 2466282, 0, 17095813, 45359632, 0, 323760077, 870624976, 0, 6367406592, 17307580710, 0, 129063054631, 353941332518, 0, 2682355470491, 7410591325928, 0, 56930627178287

Offset

1,5

Comments

The empty subset is not allowed, otherwise we would get a(2)=1. - Alois P. Heinz, Sep 03 2009

Links

Alois P. Heinz, Table of n, a(n) for n = 1..100

Formula

a(n) is the coefficient of x^3y in product(x^(-2k)+x^k(y^k+y^(-k)), k=1..n) for n>2.

Example

For n=5 we have splittings 4/23/15, 4/5/123, 13/5/24, so a(5)=3.

Maple

A113039:=proc(n) local i, j, p, t; t:= 0, 0; for j from 3 to n do p:=1; for i to j do p:=p*(x^(-2*i)+x^(i)*(y^i+y^(-i))); od; t:=t, coeff(coeff(p, x, 3), y, 1); od; t; end;
# 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:= proc(n) local m; m:= n*(n+1)/2; `if` (n>2 and irem (m, 3)=0, b(m/3-1, m/3, m/3+1, n), 0) end: seq (a(n), n=1..42); # Alois P. Heinz, Sep 03 2009

Mathematica

a[n_] := If[n <= 2, 0, Product[x^(-2k)+x^k(y^k+y^(-k)), {k, 1, n}] // SeriesCoefficient[#, {x, 0, 3}, {y, 0, 1}]&];
Table[an = a[n]; Print[n, " ", an]; an, {n, 1, 26}] (* Jean-François Alcover, Nov 17 2022 *)

Crossrefs

Cf. A112972.

Sequence in context: A318204 A306637 A111823 * A093016 A031018 A146525

Adjacent sequences: A113036 A113037 A113038 * A113040 A113041 A113042

Keyword

nonn

Author

Floor van Lamoen, Oct 12 2005

Extensions

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

Status

approved