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A092673
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a(n) = moebius(n)- moebius(n/2) where moebius(n) is zero if n is not an integer.
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10
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1, -2, -1, 1, -1, 2, -1, 0, 0, 2, -1, -1, -1, 2, 1, 0, -1, 0, -1, -1, 1, 2, -1, 0, 0, 2, 0, -1, -1, -2, -1, 0, 1, 2, 1, 0, -1, 2, 1, 0, -1, -2, -1, -1, 0, 2, -1, 0, 0, 0, 1, -1, -1, 0, 1, 0, 1, 2, -1, 1, -1, 2, 0, 0, 1, -2, -1, -1, 1, -2, -1, 0, -1, 2, 0, -1, 1, -2, -1, 0, 0, 2, -1, 1, 1, 2, 1, 0, -1, 0, 1, -1, 1, 2, 1, 0, -1, 0, 0, 0, -1, -2, -1, 0, -1, 2
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A10054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).
Setting x=1 gives us phi(n) (A000010). Setting x=0 gives A092674.
Apparently the Dirichlet inverse of A001511. - R. J. Mathar, Dec 22 2010
Given A = A115359 as an infinite lower triangular matrix and B = the Mobius sequence as a vector, A092673 = A*B. [Gary W. Adamson, (qntmpkt(AT)yahoo.com), Mar 14 2011]
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LINKS
| M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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FORMULA
| Let t(n)=binomial(n+1,2); s[1]=x; for i >= 2, s[i]=t(i)-sum_{j=1..i-1} s[j]*floor(i/j); a(n) = coefficient of x in s[n]. - Jon Perry
a(n) is multiplicative with a(2)= -2, a(4)= 1, a(2^e)= 0 if e>2. a(p)= -1, a(p^e)= 0 if e>1, p>2. - Michael Somos Mar 26 2007
a(8n)= 0 . a(2n+1)= moebius(2n+1) . a(2n)= moebius(2n)- moebius(n) . - Michael Somos Mar 26 2007
|a(n)|<=2 .
1/(1+x)= Product_{k>0} f(-x^k)^a(k) where f() is a Ramanujan theta function. - Michael Somos Mar 26 2007
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EXAMPLE
| The first few s[n] are:
x, -2*x + 3, -x + 3, x + 1, -x + 5, 2*x, -x + 7, 4, 6, 2*x + 2, -x + 11, -x + 5, -x + 13, 2*x + 4, x + 7, 8, -x + 17, 6, -x + 19, -x + 9, x + 11, 2*x + 8, -x + 23, 8, 20, 2*x + 10, 18, -x + 13, -x + 29, -2*x + 10, -x + 31, 16, x + 19.
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MATHEMATICA
| f[n_] := MoebiusMu[n] - If[OddQ@n, 0, MoebiusMu[n/2]]; Array[f, 105] (* RGWv *)
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PROG
| (PARI) s=vector(2000); t(n)=binomial(n+1, 2); s[1]=x; for(i=2, 2000, s[i]=t(i)-sum(j=1, i-1, s[j]*floor(i/j))); for(i=1, 2000, print1(", "polcoeff(s[i], 1)))
(PARI) {a(n)= if(n<1, 0, moebius(n)- if(n%2, 0, moebius(n/2)))} /* Michael Somos Mar 26 2007 */
(PARI) {a(n)= local(A, B, m); if(n<1, 0, A= x*O(x^n); B= 1+x+A; for(k=1, n, B*= eta(x^k+A)^ (m= polcoeff(B, k))); m)} /* Michael Somos Mar 26 2007 */
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CROSSREFS
| Cf. A008683 (moebius(n)), A092674, A115359.
Sequence in context: A143262 A035180 A163819 * A112400 A116861 A105242
Adjacent sequences: A092670 A092671 A092672 * A092674 A092675 A092676
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KEYWORD
| sign,mult,changed
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AUTHOR
| Jon Perry (perry(AT)globalnet.co.uk), Mar 02 2004
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