A014453 Theta series of quadratic form with Gram matrix [ 2, 0, 0; 0, 2, 1; 0, 1, 2 ].

1, 8, 12, 6, 20, 24, 0, 24, 36, 8, 24, 24, 18, 48, 24, 0, 44, 48, 12, 24, 48, 24, 48, 48, 0, 56, 24, 6, 72, 72, 24, 24, 84, 0, 24, 48, 20, 96, 48, 24, 72, 48, 0, 72, 72, 24, 48, 48, 42, 56, 60, 0, 96, 120, 0, 48, 72, 48, 72, 24, 0, 96, 72, 24, 92, 96, 24, 72, 120, 0, 48, 48, 36, 96, 72

Offset

0,2

Comments

This is the hexagonal P lattice (the even holotype) of dimension 3.

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

a(n) is the number of solutions to x^2 + y^2 + z^2 + x*y = n in integers. - Michael Somos, Jul 03 2018

Links

John Cannon, Table of n, a(n) for n = 0..5000

G. Nebe and N. J. A. Sloane, Home page for this lattice

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of a(x) * phi(x) where phi() is a Ramanujan theta function and a() is a cubic AGM theta function. - Michael Somos, May 30 2012

Expansion of (eta(q)^3 + 9 * eta(q^9)^3) * eta(q^2)^5 / (eta(q)^2 * eta(q^3) * eta(q^4)^2) in powers of q.

Convolution of A004016 and A000122. - Michael Somos, May 30 2012

Example

G.f. = 1 + 8*x + 12*x^2 + 6*x^3 + 20*x^4 + 24*x^5 + 24*x^7 + 36*x^8 + 8*x^9 + ...
G.f. = 1 + 8*q^2 + 12*q^4 + 6*q^6 + 20*q^8 + 24*q^10 + 24*q^14 + 36*q^16 + 8*q^18 + ...

Mathematica

(* A004016 *) a2[0] = 1; a2[n_] := 6*DivisorSum[n, KroneckerSymbol[#, 3]&]; (* A000122 *) a3[n_] := SeriesCoefficient[EllipticTheta[3, 0, q], {q, 0, n}]; a[n_] := Sum[a2[k]*a3[n-k], {k, 0, n}]; Table[a[n], {n, 0, 75}] (* Jean-François Alcover, Nov 04 2015, from the convolution given by Michael Somos *)
a[ n_] :=   SeriesCoefficient[ EllipticTheta[ 3, 0,  x] (QPochhammer[ x]^3 + 9 x QPochhammer[ x^9]^3) / QPochhammer[ x^3], {x, 0, n}]; (* Michael Somos, Jul 03 2018 *)

Prog

(PARI) {a(n) = if( n<1, n==0, 2 * qfrep( [ 2, 0, 0; 0, 2, 1; 0, 1, 2 ], n, 1)[n])}; /* Michael Somos, May 30 2012 */
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^3  + 9 * x * eta(x^9 + A)^3) / eta(x^3 + A) * eta(x^2 + A)^5 / (eta(x + A)^2 * eta(x^4 + A)^2), n))}; /* Michael Somos, May 30 2012 */

Crossrefs

Cf. A000122, A004016.

Sequence in context: A203836 A220665 A166173 * A160862 A152077 A215696

Adjacent sequences: A014450 A014451 A014452 * A014454 A014455 A014456

Keyword

nonn

Author

N. J. A. Sloane.

Status

approved