A186690 - OEIS

%I #41 Jun 23 2024 05:53:38

%S 1,-2,4,-4,6,-8,8,-8,13,-12,12,-16,14,-16,24,-16,18,-26,20,-24,32,-24,

%T 24,-32,31,-28,40,-32,30,-48,32,-32,48,-36,48,-52,38,-40,56,-48,42,

%U -64,44,-48,78,-48,48,-64,57,-62,72,-56,54,-80,72,-64,80,-60,60,-96,62,-64

%N Expansion of - (1/8) theta_3''(0, q) / theta_3(0, q) in powers of q.

%C If A(x) is the generating function then 1 / Pi = 8 A( exp( -Pi) ). [Plouffe, equation 1.2]

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

%D A. P. Prudnikov, Yu. A. Brychkov and O.I. Marichev, "Integrals and Series", Volume 1: "Elementary Functions", Chapter 4: "Finite Sums", New York, Gordon and Breach Science Publishers, 1986-1992, Equation (5.1.29.8).

%H Seiichi Manyama, <a href="/A186690/b186690.txt">Table of n, a(n) for n = 1..10000</a>

%H Simon Plouffe, <a href="http://arxiv.org/abs/1101.6066">Identities inspired by the Ramanujan Notebooks, Second series</a>, arXiv:1101.6066 [math.NT], 2011.

%H R. Sivaraman, José Luis López-Bonilla, and Sergio Vidal-Beltrán, <a href="http://doi.org/10.54105/ijam.A1162.103223">On the Polynomial Structure of r_k(n)</a>, Indian J. Adv. Math. (2023) Vol. 3, Iss. 2, A1162044124.

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F Multiplicative with a(2^e) = -(2^e) if e>0, a(p^e) = (p^(e+1) - 1) / (p - 1) if p > 2.

%F Expansion of (E - (1 - k^2) * K) * K / (2 Pi^2) in powers of the nome q where K, E are complete elliptic integrals.

%F Expansion of (1/2) x (d phi(x) / dx) / phi(x) in powers of x where phi() is a Ramanujan theta function.

%F G.f.: Sum_{k>0} - (-1)^k * k * x^k / (1 - x^(2*k)) = Sum_{k>0} x^(2*k-1) / (1 + x^(2*k-1))^2 = (Sum_{k>0} n^2 x^(n^2)) / (Sum_k x^(n^2)).

%F Dirichlet g.f. zeta(s) *zeta(s-1) *(1-7*2^(-s)+14*4^(-s)-8^(1-s)) / (1-2^(1-s)). - _R. J. Mathar_, Jun 01 2011

%F a(n) = -(-1)^n * A002131(n).

%F MOBIUS transform is A186111. - _Michael Somos_, Apr 25 2015

%e G.f. = q - 2*q^2 + 4*q^3 - 4*q^4 + 6*q^5 - 8*q^6 + 8*q^7 - 8*q^8 + 13*q^9 + ...

%t a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ (1/8) (EllipticE[m] - (1 - m) EllipticK[m]) EllipticK[m]/(Pi/2)^2, {q, 0, n}]];

%o (PARI) {a(n) = if( n<1, 0, -(-1)^n * sumdiv( n, d, d / gcd(d, 2)))};

%o (Python)

%o from math import prod

%o from sympy import factorint

%o def A186690(n): return (1 if n&1 else -1)*prod((p**(e+1)-1)//(p-1) if p&1 else 1<<e for p, e in factorint(n).items()) # _Chai Wah Wu_, Jun 23 2024

%Y Cf. A002131, A186111.

%K sign,look,mult

%O 1,2

%A _Michael Somos_, Feb 25 2011