A117000 a(n) = Sum_{d|n} Jacobi(2,d)*d.

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

Offset

1,3

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Inverse Mobius transform of the sequence n*A091337(n), n>=1. - R. J. Mathar, Jul 08 2011

References

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.

Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 85, Eq. (32.67).

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Michael Somos, Introduction to Ramanujan theta functions, 2019.

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Formula

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

From Michael Somos, Aug 08 2007: (Start)

Expansion of (1 - phi(q) * phi(q^2) * phi(-q)^2) / 2 in powers of q where phi() is a Ramanujan theta function.

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).

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.

G.f.: Sum_{k>0} k * x^k / (1 - x^k) * Kronecker(2, k). (End)

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

From Amiram Eldar, Jan 28 2024: (Start)

a(n) = (-1)^(n+1) * A113418(n).

Sum_{k=1..n} abs(a(k)) ~ c * n^2, where c = Pi^2/(24*sqrt(2)) = 0.290786... . (End)

Example

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 + ...

Maple

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;

Mathematica

a[n_] := Sum[JacobiSymbol[2, d]*d, {d, Divisors[n]}]; a /@ Range[80] (* Jean-François Alcover, Jan 10 2014 *)
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 *)
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 *)

Prog

(PARI) {a(n)= if( n<1, 0, sumdiv(n, d, d * kronecker(2, d)))}; /* Michael Somos, Aug 08 2007 */
(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 */
(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 */

Crossrefs

Apart from signs, same as A113418.

Cf. A091337, A111374, A117001.

Cf. A000122, A000700, A010054, A121373.

Sequence in context: A304587 A364952 A113418 * A082392 A233327 A307107

Adjacent sequences: A116997 A116998 A116999 * A117001 A117002 A117003

Keyword

sign,mult

Author

N. J. A. Sloane, Apr 15 2006

Status

approved