

A000007


The characteristic function of {0}: a(n) = 0^n.
(Formerly M0002)


1003



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, 0
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OFFSET

0,1


COMMENTS

Changing the offset to 1 gives the arithmetical function a(1) = 1, a(n) = 0 for n > 1, the identity function for Dirichlet multiplication (see Apostol).  N. J. A. Sloane
Changing the offset to 1 makes this the decimal expansion of 1.  N. J. A. Sloane, Nov 13 2014
Hankel transform (see A001906 for definition) of A000007 (powers of 0), A000012 (powers of 1), A000079 (powers of 2), A000244 (powers of 3), A000302 (powers of 4), A000351 (powers of 5), A000400 (powers of 6), A000420 (powers of 7), A001018 (powers of 8), A001019 (powers of 9), A011557 (powers of 10), A001020 (powers of 11), etc.  Philippe Deléham, Jul 07 2005
This is the identity sequence with respect to convolution.  David W. Wilson, Oct 30 2006
The alternating sum of the nth row of Pascal's triangle gives the characteristic function of 0, a(n) = 0^n.  Daniel Forgues, May 25 2010
The number of maximal selfavoiding walks from the NW to SW corners of a 1 X n grid.  Sean A. Irvine, Nov 19 2010
Historically there has been some disagreement as to whether 0^0 = 1. Graphing x^0 seems to support that conclusion, but graphing 0^x instead suggests that 0^0 = 0. Euler and Knuth have argued in favor of 0^0 = 1. For some calculators, 0^0 triggers an error, while in Mathematica, 0^0 is Indeterminate.  Alonso del Arte, Nov 15 2011
Another consequence of changing the offset to 1 is that then this sequence can be described as the sum of Moebius mu(d) for the divisors d of n.  Alonso del Arte, Nov 28 2011
With the convention 0^0 = 1, 0^n = 0 for n > 0, the sequence a(n) = 0^nk, which equals 1 when n = k and is 0 for n >= 0, has g.f. x^k. A000007 is the case k = 0.  George F. Johnson, Mar 08 2013
A fixed point of the run length transform.  Chai Wah Wu, Oct 21 2016


REFERENCES

T. M. Apostol, Introduction to Analytic Number Theory, SpringerVerlag, 1976, page 30.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.


LINKS



FORMULA

As a function of Bernoulli numbers (cf. A027641: (1, 1/2, 1/6, 0, 1/30, ...)), triangle A074909 (the beheaded Pascal's triangle) * B_n as a vector = [1, 0, 0, 0, 0, ...].  Gary W. Adamson, Mar 05 2012
a(n) = Sum_{k = 0..n} exp(2*Pi*i*k/(n+1)) is the sum of the (n+1)th roots of unity.  Franz Vrabec, Nov 09 2012
Inverse binomial transform of A000012. (End)


MAPLE

A000007 := proc(n) if n = 0 then 1 else 0 fi end: seq(A000007(n), n=0..20);
spec := [A, {A=Z} ]: seq(combstruct[count](spec, size=n+1), n=0..20);


MATHEMATICA

Table[If[n == 0, 1, 0], {n, 0, 99}]
Table[Boole[n == 0], {n, 0, 99}] (* Michael Somos, Aug 25 2012 *)
Join[{1}, LinearRecurrence[{1}, {0}, 102]] (* Ray Chandler, Jul 30 2015 *)


PROG

(PARI) {a(n) = !n};
(Magma) [1] cat [0:n in [1..100]]; // Sergei Haller, Dec 21 2006
(Haskell)
a000007 = (0 ^)
a000007_list = 1 : repeat 0
(Python)


CROSSREFS



KEYWORD



AUTHOR



STATUS

approved



