A005883 Theta series of square lattice with respect to deep hole.

4, 8, 4, 8, 8, 0, 12, 8, 0, 8, 8, 8, 4, 8, 0, 8, 16, 0, 8, 0, 4, 16, 8, 0, 8, 8, 0, 8, 8, 8, 4, 16, 0, 0, 8, 0, 16, 8, 8, 8, 0, 0, 12, 8, 0, 8, 16, 0, 8, 8, 0, 16, 0, 0, 0, 16, 12, 8, 8, 0, 8, 8, 0, 0, 8, 8, 16, 8, 0, 8, 8, 0, 12, 8, 0, 0, 16, 0, 8, 8, 0, 24, 0, 8, 8, 0, 0, 8, 8, 0, 4, 16, 8, 8, 16, 0, 0

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

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

In [Jacobi 1829] on page 105 is equation 18: "2 k K / Pi = 4 sqrt(q) + 8 sqrt(q^5) + 4 sqrt(q^9) [...]". - Michael Somos, Sep 09 2012

Examples

			4 + 8*x + 4*x^2 + 8*x^3 + 8*x^4 + 12*x^6 + 8*x^7 + 8*x^9 + 8*x^10 + 8*x^11 + ...
4*q + 8*q^3 + 4*q^5 + 8*q^7 + 8*q^9 + 12*q^13 + 8*q^15 + 8*q^19 + 8*q^21 + ...
Theta = 4*q^(1/2) + 8*q^(5/2) + 4*q^(9/2) + 8*q^(13/2) + 8*q^(17/2) + ...
		

References

• J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 106.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
• C. G. J. Jacobi, Fundamenta Nova Theoriae Functionum Ellipticarum, 1829.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A008441.

Programs

Mathematica

a[0] = 4; a[n_] := 4*DivisorSum[4n+1, (-1)^Quotient[#, 2]&]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Nov 04 2015, translated from PARI *)
s = 4*(QPochhammer[q^2]^4/QPochhammer[q]^2)+O[q]^100; CoefficientList[s, q] (* Jean-François Alcover, Nov 09 2015 *)

PARI

{a(n) = if( n<0, 0, n = 4*n + 1; 4 * sumdiv(n, d, (-1)^(d\2)))} /* Michael Somos, Oct 31 2006 */

Formula

Expansion of Jacobi theta constant q^(-1/2) * theta_2(q)^2 in powers of q^2. - Michael Somos, Oct 31 2006

G.f.: 4 * (Product_{k>0} (1 - x^k) * (1 + x^(2*k))^2)^2. - Michael Somos, Oct 31 2006

From Michael Somos, Sep 09 2012: (Start)

Expansion of 4 * psi(x)^2 in powers of x where psi() is a Ramanujan theta function.

Expansion of q^(-1) * (1/2) * (1 - k') * K / (Pi/2) in powers of q^4 where k', K are Jacobi elliptic functions.

Expansion of q^(-1/2) * k * K / (Pi/2) in powers of q^2 where k, K are Jacobi elliptic functions.

Expansion of q^(-1/4) * 2 * k^(1/2) * K / (Pi/2) in powers of q where k, K are Jacobi elliptic functions.

Expansion of 4 * q^(-1/4) * eta(q^2)^4 / eta(q)^2 in powers of q.

a(n) = 4 * A008441(n). (End)