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%I #10 Jan 24 2019 16:16:25
%S 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,
%T 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,
%U 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
%N a(1) = -1, otherwise a(n) = 0.
%C a(0) = 0; for n >= 1, a(n) = function of negative sign for Dirichlet convolution.
%C 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).
%C a(n) = the sum of the cranks of all partitions of n. - _Michael Somos_, Nov 10 2013
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Crank_of_a_partition">Crank of a partition</a>
%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (1).
%F G.f.: -x.
%F 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.
%t a[ n_] := -Boole[n == 1] (* _Michael Somos_, Nov 10 2013 *)
%t PadRight[{0,-1},120,0] (* _Harvey P. Dale_, Jan 24 2019 *)
%o (PARI) {a(n) = -(n == 1)} /* _Michael Somos_, Nov 10 2013 */
%Y Cf. A000004, A063524, A000007, A000012, A057428.
%K sign
%O 0,1
%A _Jaroslav Krizek_, Apr 04 2009
%E Edited by _N. J. A. Sloane_, Apr 09 2009
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