|
%I #36 Jan 28 2024 02:00:44
%S 1,1,-2,1,-4,-2,8,1,7,-4,-10,-2,-12,8,8,1,18,7,-18,-4,-16,-10,24,-2,
%T 21,-12,-20,8,-28,8,32,1,20,18,-32,7,-36,-18,24,-4,42,-16,-42,-10,-28,
%U 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
%N a(n) = Sum_{d|n} Jacobi(2,d)*d.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C Inverse Mobius transform of the sequence n*A091337(n), n>=1. - _R. J. Mathar_, Jul 08 2011
%D Henry J. S. Smith, Report on the Theory of Numbers, reprinted in Vol. 1 of his Collected Math. Papers, Chelsea, NY, 1979, see p. 323.
%D Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 85, Eq. (32.67).
%H G. C. Greubel, <a href="/A117000/b117000.txt">Table of n, a(n) for n = 1..1000</a>
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>, 2019.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F 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_, Apr 15 2006
%F From _Michael Somos_, Aug 08 2007: (Start)
%F Expansion of (1 - phi(q) * phi(q^2) * phi(-q)^2) / 2 in powers of q where phi() is a Ramanujan theta function.
%F 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).
%F 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.
%F G.f.: Sum_{k>0} k * x^k / (1 - x^k) * Kronecker(2, k). (End)
%F Logarithmic derivative of A111374, the reciprocal of the Goellnitz-Gordon continued fraction: 1+x + x^2/(1+x^3 + x^4/(1+x^5 + x^6/(1+x^7 +...))) in powers of x. - _Paul D. Hanna_, Jan 10 2014
%F From _Amiram Eldar_, Jan 28 2024: (Start)
%F a(n) = (-1)^(n+1) * A113418(n).
%F Sum_{k=1..n} abs(a(k)) ~ c * n^2, where c = Pi^2/(24*sqrt(2)) = 0.290786... . (End)
%e G.f. = q + q^2 - 2*q^3 + q^4 - 4*q^5 - 2*q^6 + 8*q^7 + q^8 + 7*q^9 - 4*q^10 - 10*q^11 + ...
%p 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;
%t a[n_] := Sum[JacobiSymbol[2, d]*d, {d, Divisors[n]}]; a /@ Range[80] (* _Jean-François Alcover_, Jan 10 2014 *)
%t a[ n_] := SeriesCoefficient[ (1 - EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^2] EllipticTheta[ 4, 0, q]^2) / 2, {q, 0, n}]; (* _Michael Somos_, Apr 26 2015 *)
%t a[ n_] := SeriesCoefficient[ (1 - QPochhammer[ q]^2 QPochhammer[ q^2] QPochhammer[ q^4]^3 / QPochhammer[ q^8]^2) / 2, {q, 0, n}]; (* _Michael Somos_, Apr 26 2015 *)
%o (PARI) {a(n)= if( n<1, 0, sumdiv(n, d, d * kronecker(2, d)))}; /* _Michael Somos_, Aug 08 2007 */
%o (PARI) {a(n) = my(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 */
%o (PARI) {a(n) = my(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 */
%Y Apart from signs, same as A113418.
%Y Cf. A091337, A111374, A117001.
%Y Cf. A000122, A000700, A010054, A121373.
%K sign,mult
%O 1,3
%A _N. J. A. Sloane_, Apr 15 2006
|
