A186111 -id:A186111 - cp's OEIS frontend

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

Michael Somos, Feb 25 2011

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).

			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 + ...

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).

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

Cf. A002131, A186111.

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

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

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<Chai Wah Wu, Jun 23 2024

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

A186813 a(n) = n if n odd, a(2n) = 3n if n odd, a(4n) = 2n.

Original entry on oeis.org

0, 1, 3, 3, 2, 5, 9, 7, 4, 9, 15, 11, 6, 13, 21, 15, 8, 17, 27, 19, 10, 21, 33, 23, 12, 25, 39, 27, 14, 29, 45, 31, 16, 33, 51, 35, 18, 37, 57, 39, 20, 41, 63, 43, 22, 45, 69, 47, 24, 49, 75, 51, 26, 53, 81, 55, 28, 57, 87, 59, 30, 61, 93, 63, 32, 65, 99, 67, 34, 69, 105, 71, 36

Offset: 0

Michael Somos, Feb 27 2011

			G.f. = x + 3*x^2 + 3*x^3 + 2*x^4 + 5*x^5 + 9*x^6 + 7*x^7 + 4*x^8 + 9*x^9 + ...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Michael Somos, Rational Function Multiplicative Coefficients
Index entries for linear recurrences with constant coefficients, signature (2,-3,4,-3,2,-1).

Cf. A068073, A134172, A186111.

Cf. A187601. - Bruno Berselli, Mar 12 2011

m:=25; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(x*(1+x)*(1+x^3)/((1-x)*(1+x^2))^2)); // G. C. Greubel, Aug 14 2018

CoefficientList[Series[x(1+x)(1+x^3)/((1-x)(1+x^2))^2,{x,0,80}],x] (* Harvey P. Dale, Mar 06 2011 *)
a[ n_] := n/2 {2, 3, 2, 1}[[ Mod[ n, 4, 1]]]; (* Michael Somos, May 04 2015 *)

PARI
```
{a(n) = n/2 * [1, 2, 3, 2][n%4 + 1]};
```

{a(n) = sign(n) * polcoeff( x * (1 + x) * (1 + x^3) / ((1 - x) * (1 + x^2))^2 + x * O(x^abs(n)), abs(n))};

a(n) is multiplicative with a(2) = 3, a(2^e) = 2^(e-1) if e>1, a(p^e) = p^e if p>2.
Euler transform of length 6 sequence [3, -3, 1, 2, 0, -1].
G.f.: x * (1 + x) * (1 + x^3) / ((1 - x) * (1 + x^2))^2.
G.f.: x * (1 - x^2)^3 * (1 - x^6) / ((1 - x)^3 * (1 - x^3) * (1 - x^4)^2). - Michael Somos, May 04 2015
G.f.: f(x) - f(-x^2) where f(x) := x/(1-x)^2. - Michael Somos, May 04 2015
a(n) = -a(-n) for all n in Z. a(n) = n/2 * A068073(n).
a(n) = n*(4-i^n-(-i)^n)/4 with i=sqrt(-1). - Bruno Berselli, Mar 10 2011
a(n) = A134172(n) + A134172(n+1). - Michael Somos, May 04 2015
a(n) = -(-1)^n * A186111(n). - Michael Somos, May 07 2015
a(n) = n - n*cos(n*Pi/2)/2. - Wesley Ivan Hurt, May 05 2021
Dirichlet g.f.: zeta(s-1) * (1 + 1/2^s - 1/4^(s-1)). - Amiram Eldar, Oct 26 2023

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

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A186813 a(n) = n if n odd, a(2n) = 3n if n odd, a(4n) = 2n.

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