A159075 a(1) = -1, otherwise a(n) = 0.

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Offset: 0

Author

Jaroslav Krizek, Apr 04 2009

Keywords

• sign

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).

Crossrefs

Cf. A000004, A063524, A000007, A000012, A057428.

Programs

Mathematica

a[ n_] := -Boole[n == 1] (* Michael Somos, Nov 10 2013 *)
PadRight[{0,-1},120,0] (* Harvey P. Dale, Jan 24 2019 *)

PARI

{a(n) = -(n == 1)} /* Michael Somos, Nov 10 2013 */

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.

Extensions

Edited by N. J. A. Sloane, Apr 09 2009