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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)
16
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

REFERENCES

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 294.

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[0] = 2; Table[a[n], {n, 0, 24}](* Jean-Fran├žois 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

N. J. A. Sloane

EXTENSIONS

More terms from Michael Somos, Jun 10 2000

STATUS

approved

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Last modified February 25 12:38 EST 2018. Contains 299654 sequences. (Running on oeis4.)