A014453 - OEIS

%I #25 Mar 12 2021 22:24:41

%S 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,

%T 56,24,6,72,72,24,24,84,0,24,48,20,96,48,24,72,48,0,72,72,24,48,48,42,

%U 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

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

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

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

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

%H John Cannon, <a href="/A014453/b014453.txt">Table of n, a(n) for n = 0..5000</a>

%H G. Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/hexagonalPe.html">Home page for this lattice</a>

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F 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

%F 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.

%F Convolution of A004016 and A000122. - _Michael Somos_, May 30 2012

%e 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 + ...

%e 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 + ...

%t (* 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_ *)

%t 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 *)

%o (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 */

%o (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 */

%Y Cf. A000122, A004016.

%K nonn

%O 0,2

%A _N. J. A. Sloane_.