A005883 Theta series of square lattice with respect to deep hole. (Formerly M3319)

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

0,1

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

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

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)

Example

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) + ...

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 *)

Prog

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

Crossrefs

Cf. A008441.

Sequence in context: A348573 A010713 A105398 * A055026 A336818 A205681

Adjacent sequences: A005880 A005881 A005882 * A005884 A005885 A005886

Keyword

nonn

Author

N. J. A. Sloane

Status

approved