A186690 Expansion of - (1/8) theta_3''(0, q) / theta_3(0, q) in powers of q.

1, -2, 4, -4, 6, -8, 8, -8, 13, -12, 12, -16, 14, -16, 24, -16, 18, -26, 20, -24, 32, -24, 24, -32, 31, -28, 40, -32, 30, -48, 32, -32, 48, -36, 48, -52, 38, -40, 56, -48, 42, -64, 44, -48, 78, -48, 48, -64, 57, -62, 72, -56, 54, -80, 72, -64, 80, -60, 60, -96, 62, -64

Offset

1,2

Comments

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

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

References

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).

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Simon Plouffe, Identities inspired by the Ramanujan Notebooks, Second series, arXiv:1101.6066 [math.NT], 2011.

R. Sivaraman, José Luis López-Bonilla, and Sergio Vidal-Beltrán, On the Polynomial Structure of r_k(n), Indian J. Adv. Math. (2023) Vol. 3, Iss. 2, A1162044124.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

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

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

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

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)).

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

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

MOBIUS transform is A186111. - Michael Somos, Apr 25 2015

Example

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 + ...

Mathematica

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

Prog

(PARI) {a(n) = if( n<1, 0, -(-1)^n * sumdiv( n, d, d / gcd(d, 2)))};
(Python)
from math import prod
from sympy import factorint
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

Crossrefs

Cf. A002131, A186111.

Sequence in context: A288772 A053196 A159634 * A002131 A230641 A373800

Adjacent sequences: A186687 A186688 A186689 * A186691 A186692 A186693

Keyword

sign,look,mult

Author

Michael Somos, Feb 25 2011

Status

approved