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A113039 Number of ways the set {1,2,...,n} can be split into three subsets of which the three sums are consecutive. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,5
COMMENTS
The empty subset is not allowed, otherwise we would get a(2)=1. - Alois P. Heinz, Sep 03 2009
LINKS
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
KEYWORD
nonn
AUTHOR
Floor van Lamoen, Oct 12 2005
EXTENSIONS
Extended beyond a(25) by Alois P. Heinz, Sep 03 2009
STATUS
approved

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Last modified June 20 06:40 EDT 2024. Contains 373512 sequences. (Running on oeis4.)