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 A159075 a(1) = -1, otherwise a(n) = 0. 0
 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS a(0) = 0; for n >= 1, a(n) = function of negative sign for Dirichlet convolution. a(n) = Dirichlet inverse of itself. a(n) * 0(n) = a(n) * A000004(n) = 0(n) = A000004(n), a(n) * b(n) = -[b(n)], a(n) * a(n) = A063524(n) = A000007(n - 1) for n >= 1 (identity function for Dirichlet convolution), where operation * denotes Dirichlet convolution for n >= 1, b(n) is any function. Dirichlet convolution of functions a(n), b(n) is function c(n) = a(n) * b(n) = Sum_{d|n} a(d)*b(n/d). a(n) = the sum of the cranks of all partitions of n. - Michael Somos, Nov 10 2013 LINKS Wikipedia, Crank of a partition Index entries for linear recurrences with constant coefficients, signature (1). FORMULA G.f.: -x. Sum_{d|n} a(d)*a(n/d) = Sum_{1<=k<=n} a(k)*a(n-k+1) = A063524(n) = A000007(n - 1) for n >= 1. Sum_{d|n} a(d)*a(d) = Sum_{1<=k<=n} a(k)*a(k) = A000012(n) for n >= 1. Sum_{d|n} a(d)*b(n/d) = Sum_{1<=k<=n} a(k)*b(n-k+1) = -[b(n)] for any function b(n) and n >= 1. Sum_{d|n} a(d)*b(d) = Sum_{1<=k<=n} a(k)*b(k) = A057428(n) for any function b(n) with Abs[b(1)] >= 1 and n >= 1. a(n) = (-1) * A063524(n). a(n) = (-1) * A000007(n - 1) for n >= 1. Abs[a(n)] = A063524(n). Abs[a(n)] = A000007(n - 1) for n >= 1. MATHEMATICA a[ n_] := -Boole[n == 1] (* Michael Somos, Nov 10 2013 *) PadRight[{0, -1}, 120, 0] (* Harvey P. Dale, Jan 24 2019 *) PROG (PARI) {a(n) = -(n == 1)} /* Michael Somos, Nov 10 2013 */ CROSSREFS Cf. A000004, A063524, A000007, A000012, A057428. Sequence in context: A257834 A173950 A157928 * A178333 A063524 A326168 Adjacent sequences:  A159072 A159073 A159074 * A159076 A159077 A159078 KEYWORD sign AUTHOR Jaroslav Krizek, Apr 04 2009 EXTENSIONS Edited by N. J. A. Sloane, Apr 09 2009 STATUS approved

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Last modified November 30 05:31 EST 2020. Contains 338781 sequences. (Running on oeis4.)