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Search: id:A117000
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| A117000 |
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Sum_{d|n} Jacobi(2,d)*d. |
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+0
9
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| 1, 1, -2, 1, -4, -2, 8, 1, 7, -4, -10, -2, -12, 8, 8, 1, 18, 7, -18, -4, -16, -10, 24, -2, 21, -12, -20, 8, -28, 8, 32, 1, 20, 18, -32, 7, -36, -18, 24, -4, 42, -16, -42, -10, -28, 24, 48, -2, 57, 21, -36, -12, -52, -20, 40, 8, 36, -28, -58, 8, -60, 32, 56, 1, 48, 20, -66, 18, -48, -32, 72, 7, 74, -36, -42, -18, -80, 24, 80, -4
(list; graph; listen)
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OFFSET
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1,3
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REFERENCES
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H. J. S. Smith, Report on the Theory of Numbers, reprinted in Vol. 1 of his Collected Math. Papers, Chelsea, NY, 1979, see p. 323.
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 85, Eq. (32.67).
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FORMULA
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G.f.: Sum_{k>0} x^k*(1+x^(2*k))*(1-4*x^(2*k)+x^(4*k))/(1+x^(4*k))^2. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Apr 15 2006
Expansion of (1- phi(q)* phi(q^2)* phi(-q)^2)/2 in powers of q where phi() is a Ramanujan theta function. - Michael Somos Aug 08 2007
a(n) is multiplicative with a(2^e) = 1, a(p^e) = (p^(e+1)-1)/(p-1) if p == 1, 7 (mod 8), ((-p)^(e+1)-1)/(-p-1) if p == 3, 5 (mod 8). - Michael Somos Aug 08 2007
Given g.f. A(x), then B(x)= 1-2*A(x) satisfies 0= f(B(x), B(x^2), B(x^4)) where f(u, v, w)= v^4 +u^2*v^2 +2*u^2*w^2 +2*u*v*w* (-u+2*v-2*w) -2*u*v^3. - Michael Somos Aug 08 2007
G.f.: Sum_{k>0} k* x^k/(1-x^k)* kronecker(2, k). - Michael Somos Aug 08 2007
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MAPLE
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with(numtheory); A117000:=proc(n) local d, t1, t2; t1:=0; t2:=0; for d from 1 to n do if n mod d = 0 then t1:=t1+jacobi(2, d)*d; fi; od: t1; end;
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PROGRAM
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(PARI) {a(n)= if(n<1, 0, sumdiv(n, d, d* kronecker(2, d)))} /* Michael Somos Aug 08 2007 */
(PARI) {a(n)= local(A, p, e); if(n<1, 0, A=factor(n); prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==2, 1, if(abs(p%8-4)==3, (p^(e+1)-1)/(p-1), ((-p)^(e+1)-1)/(-p-1))))))} /* Michael Somos Aug 08 2007 */
(PARI) {a(n)= local(A); if(n<0, 0, A= x*O(x^n); polcoeff( (1-eta(x+A)^2* eta(x^2+A)* eta(x^4+A)^3/ eta(x^8+A)^2)/2, n))} /* Michael Somos Aug 08 2007 */
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CROSSREFS
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Apart from signs, same as A113418. Cf. A117001.
Sequence in context: A079966 A101707 A113418 this_sequence A082392 A085086 A137206
Adjacent sequences: A116997 A116998 A116999 this_sequence A117001 A117002 A117003
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KEYWORD
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sign,mult
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AUTHOR
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njas, Apr 15 2006
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