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A000980
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Number of ways of writing 0 as Sum_{k=-n..n} e(k)*k, where e(k) is 0 or 1.
(Formerly M1155 N0439)
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31
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2, 4, 8, 20, 52, 152, 472, 1520, 5044, 17112, 59008, 206260, 729096, 2601640, 9358944, 33904324, 123580884, 452902072, 1667837680, 6168510256, 22903260088, 85338450344, 318995297200, 1195901750512, 4495448217544, 16940411201280, 63983233268592
(list;
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history;
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OFFSET
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0,1
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COMMENTS
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The 4-term sequence 2,4,8,20 is the answer to the "Solitaire Army" problem, or checker-jumping puzzle. It is too short to have its own entry. See Conway et a;., Winning Ways, Vol. 2, pp. 715-717. - N. J. A. Sloane, Mar 01 2018
Number of subsets of {-n..n} with sum 0. Also the number of subsets of {0..2n} that are empty or have mean n. For median instead of mean we have twice A024718. - Gus Wiseman, Apr 23 2023
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REFERENCES
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L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 294.
E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see pp. 715-717.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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Constant term of Product_{k=-n..n} (1+x^k).
Coefficient of x^(n*(n+1)/2) in 2*Product_{k=1..n} (1+x^k)^2. - Sean A. Irvine, Oct 03 2011
(End)
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EXAMPLE
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The a(0) = 2 through a(2) = 8 subsets of {-n..n} with sum 0 are:
{} {} {}
{0} {0} {0}
{-1,1} {-1,1}
{-1,0,1} {-2,2}
{-1,0,1}
{-2,0,2}
{-2,-1,1,2}
{-2,-1,0,1,2}
(End)
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MAPLE
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b:= proc(n, i) option remember; `if`(n>i*(i+1)/2, 0,
`if`(i=0, 1, 2*b(n, i-1)+b(n+i, i-1)+b(abs(n-i), i-1)))
end:
a:=n-> 2*b(0, n):
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MATHEMATICA
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a[n_] := SeriesCoefficient[ Product[1+x^k, {k, -n, n}], {x, 0, 0}]; a[0] = 2; Table[a[n], {n, 0, 24}](* Jean-François Alcover, Nov 28 2011 *)
nmax = 26; d = {2}; a1 = {};
Do[
i = Ceiling[Length[d]/2];
AppendTo[a1, If[i > Length[d], 0, d[[i]]]];
d = PadLeft[d, Length[d] + 2 n] + PadRight[d, Length[d] + 2 n] +
2 PadLeft[PadRight[d, Length[d] + n], Length[d] + 2 n];
, {n, nmax}];
Table[Length[Select[Subsets[Range[-n, n]], Total[#]==0&]], {n, 0, 5}] (* Gus Wiseman, Apr 23 2023 *)
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PROG
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(PARI) a(n)=polcoeff(prod(k=-n, n, 1+x^k), 0)
(Haskell) a000980 n = length $ filter ((== 0) . sum) $ subsequences [-n..n]
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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