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 A000980 Number of ways of writing 0 as Sum_{k=-n..n} e(k)*k, where e(k) is 0 or 1. (Formerly M1155 N0439) 23
 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; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS 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 REFERENCES 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). LINKS T. D. Noe, Alois P. Heinz, and Ray Chandler, Table of n, a(n) for n = 0..1668 (terms < 10^1000, first 201 terms from T. D. Noe, next 200 terms from Alois P. Heinz, next 1268 terms from Ray Chandler) Eunice Y. S. Chan, R. M. Corless, Narayana, Mandelbrot, and A New Kind of Companion Matrix, arXiv preprint arXiv:1606.09132 [math.CO], 2016. R. C. Entringer, Representation of m as Sum_{k=-n..n} epsilon_k k, Canad. Math. Bull., 11 (1968), 289-293. Steven R. Finch, Signum equations and extremal coefficients, February 7, 2009. [Cached copy, with permission of the author] J. H. van Lint, Representations of 0 as Sum_{k = -N..N} epsilon_k*k, Proc. Amer. Math. Soc., 18 (1967), 182-184. FORMULA Constant term of Product_{k=-n..n} (1+x^k). a(n) = sum_i A067059(2n+1-i, i) = 2+2*sum_j A047997(n, j); i.e., sum of alternate antidiagonals of A067059 and two more than twice row sums of A047997. - Henry Bottomley, Aug 11 2002 a(n) = A004171(n) - 2*A181765(n). Coefficient of x^(n*(n+1)/2) in 2*prod(k=1..n,(1+x^k)^2). - Sean A. Irvine, Oct 03 2011 MAPLE 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): seq(a(n), n=0..40); # Alois P. Heinz, Mar 10 2014 MATHEMATICA a[n_] := SeriesCoefficient[ Product[1+x^k, {k, -n, n}], {x, 0, 0}]; a = 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}]; a1 (* Ray Chandler, Mar 15 2014 *) PROG (PARI) a(n)=polcoeff(prod(k=-n, n, 1+x^k), 0) (Haskell) a000980 n = length \$ filter ((== 0) . sum) \$ subsequences [-n..n] CROSSREFS A047653(n) = a(n)/2. Bisection of A084239. Cf. A063865, A141000. Sequence in context: A218088 A222320 A089976 * A123611 A082279 A113180 Adjacent sequences:  A000977 A000978 A000979 * A000981 A000982 A000983 KEYWORD nonn,nice AUTHOR EXTENSIONS More terms from Michael Somos, Jun 10 2000 STATUS approved

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Last modified June 26 04:11 EDT 2019. Contains 324369 sequences. (Running on oeis4.)