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A000007
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The characteristic function of 0: a(n) = 0^n.
(Formerly M0002)
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291
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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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). - njas
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 DELEHAM, Jul 07 2005
This is the identity sequence with respect to convolution. - David W. Wilson, Oct 30 2006
a(A000004(n)) = 1; a(A000027(n)) = 0. [From Reinhard Zumkeller, Oct 12 2008]
The alternating sum of the nth row of Pascal's triangle gives the characteristic function of 0, a(n) = 0^n. [From Daniel Forgues, May 25 2010]
The number of maximal self-avoiding walks from the NW to SW corners of an 1-by-n grid. [From 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. [From 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
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REFERENCES
| T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 30.
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| David Wasserman, Table of n, a(n) for n = 0..1000
Dr. Math, 0^0 (zero to the zero power)
Daniele A. Gewurz and Francesca Merola, Sequences realized as Parker vectors ..., J. Integer Seqs., Vol. 6, 2003.
Donald E. Knuth, Two notes on notation. See page 6 on 0^0.
Index entries for "core" sequences
Index entries for characteristic functions
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FORMULA
| Multiplicative with a(p^e) = 0. - David W. Wilson, Sep 01, 2001
a(n)= floor(1/(n+1)). - Franz Vrabec, Aug 24 2005
a(n)=((n+1)!^2 mod (n+2))*((n+2)!^2 mod (n+3)), with n>=0 - Paolo P. Lava, Apr 24 2007
a(n)=1-{[(n+1)!+1] mod (n+1)}, with n>=0. - Paolo P. Lava, May 22 2007
a(n)=1-[(n+2) mod (n+1)], with n>=0. - Paolo P. Lava, Jun 27 2007
a(n)=C(2*n,n) mod 2 - Paolo P. Lava, Aug 31 2007
a(n)=((-1)^A000040(n)+1)/2. [From Juri-Stepan Gerasimov, Oct 25 2009]
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MAPLE
| A000007 := proc(n) if n = 0 then 1 else 0; fi; end;
with(combstruct); spec := [A, {A=Z} ]; [seq(combstruct[count](spec, size=n), n=1..20)];
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MATHEMATICA
| Table[If[n == 0, 1, 0], {n, 0, 99}]
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PROG
| (PARI) a(n)=!n; for(n=0, 100, print1(a(n)", "))
(MAGMA) [1] cat [0:n in [1..100]]; - from Sergei Haller, Dec 21 2006
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CROSSREFS
| Characteristic function of g: this sequence (g=0), A063524 (g=1), A185012 (g=2), A185013 (g=3), A185014 (g=4), A185015 (g=5), A185016 (g=6), A185017 (g=7). - Jason Kimberley, Oct 14 2011
Characteristic function of multiples of g: this sequence (g=0), A000012 (g=1), A059841 (g=2), A079978 (g=3), A121262 (g=4), A079998 (g=5), A079979 (g=6), A082784 (g=7). - Jason Kimberley, Oct 14 2011
Sequence in context: A185013 A185012 A185017 * A014041 A015868 A015824
Adjacent sequences: A000004 A000005 A000006 * A000008 A000009 A000010
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KEYWORD
| core,easy,nonn,mult
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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