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%I #41 Dec 29 2023 03:37:48
%S 1,6,6,42,6,36,42,48,6,150,36,72,42,84,48,252,6,108,150,120,36,336,72,
%T 144,42,186,84,474,48,180,252,192,6,504,108,288,150,228,120,588,36,
%U 252,336,264,72,900,144,288,42,342,186,756,84,324,474,432,48,840,180,360,252,372
%N Theta series of 4-dimensional lattice QQF.4.g.
%C This sequence is obtainable, from eta products, by expanding the quotient of Eq. (135) over Eq. (105) in Broadhurst (arXiv:1604.03057). See PARI program below. - _David Broadhurst_, Apr 12 2016
%C Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%H John Cannon, <a href="/A125510/b125510.txt">Table of n, a(n) for n = 0..5000</a>
%H David Broadhurst, <a href="http://arxiv.org/abs/1604.03057">Feynman integrals, L-series and Kloosterman moments</a>, arXiv:1604.03057 [physics.gen-ph], 2016.
%H G. Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/QQF.4.g.html">Home page for this lattice</a>
%F Expansion of a(x) * a(x^2) in powers of x where a() is a cubic AGM theta function. - _Michael Somos_, Feb 10 2011
%F G.f.: 1 + 6 * (Sum_{k>0} F(x^k) + 3 * F(x^(3*k))) where F(x) = (x + x^3) / (1 - x^2)^2. - _Michael Somos_, Feb 10 2011
%F G.f.: 1 + 6 * (Sum_{k>0} k * F(x^k) + (3*k) * F(x^(3*k)))) where F(x) = x / (1 + x). - _Michael Somos_, Feb 10 2011
%F a(n) = 6*b(n) where b() is multiplicative with b(2^e) = 1, b(3^e) = 3^(e+1) - 2, b(p^e) = (p^(e+1) - 1) / (p-1) if p>3. - _Michael Somos_, Feb 17 2017
%F Expansion of (eta(q) * eta(q^2))^4 + 9 * (eta(q^3) * eta(q^6))^4) / (eta(q) * eta(q^2) * eta(q^3) * eta(q^6)) in powers of q. - _Michael Somos_, Feb 17 2017
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = 6 (t/i)^2 f(t) where q = exp(2 Pi i t). - _Michael Somos_, Feb 17 2017
%F G.f. A(x) = (F(x) + 3*F(x^3)) / 4 where F() = g.f. of A004011. - _Michael Somos_, Feb 17 2017
%F a(n) = A282544(2*n). - _Michael Somos_, Feb 18 2017
%F Sum_{k=1..n} a(k) ~ Pi^2 * n^2 / 3. - _Vaclav Kotesovec_, Dec 29 2023
%e G.f. = 1 + 6*x + 6*x^2 + 42*x^3 + 6*x^4 + 36*x^5 + 42*x^6 + 48*x^7 + 6*x^8 + ...
%e G.f. = 1 + 6*q^2 + 6*q^4 + 42*q^6 + 6*q^8 + 36*q^10 + 42*q^12 + 48*q^14 + 6*q^16 + ...
%t a[n_] := 6*(DivisorSum[n, Mod[#, 2]*# &] + If[Mod[n, 3] != 0, 0, 3 * DivisorSum[n/3, Mod[#, 2]*# &]]); a[0]=1; Table[a[n], {n, 0, 70}] (* _Jean-François Alcover_, Dec 02 2015, adapted from PARI *)
%t a[ n_] := If[ n < 1, Boole[n == 0], 6 Times @@ (Which[# < 3, 1, # == 3, 3^(#2 + 1) - 2, True, (#^(#2 + 1) - 1) / (# - 1)] & @@@ FactorInteger@n)]; (* _Michael Somos_, Feb 17 2017 *)
%o (PARI) {a(n) = if( n<1, n==0, 6 * (sumdiv( n, d, (d%2) * d) + if( n%3, 0, 3 * sumdiv( n/3, d, (d%2) * d))))}; /* _Michael Somos_, Feb 10 2011 */
%o (PARI) {et(n)=eta(q^n+O(q^(nt+1)));}
%o {nt=5000;et16=et(1)*et(6);et23=et(2)*et(3);
%o Eq105=(et16*et23)^2;
%o Eq135=(et23^3/et16)^3+q*(et16^3/et23)^3;
%o ans=Vec(Eq135/Eq105);
%o for(n=0,nt,print(n" "ans[n+1]));} /* _David Broadhurst_, Apr 12 2016 */
%o (PARI) {a(n) = if( n<1, n==0, my(A, p, e); A = factor(n); 6 * prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 1, p==3, 3^(e+1) - 2, (p^(e+1) - 1) / (p - 1))))}; /* _Michael Somos_, Feb 17 2017 */
%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^2 + A)^3 + 9*x^2*eta(x^18 + A)^3) / (eta(x^3 + A) * eta(x^6 + A)), n))}; /* _Michael Somos_, Feb 17 2017 */
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( ((eta(x + A) * eta(x^2 + A))^4 + 9*x* (eta(x^3 + A) * eta(x^6 + A))^4) / (eta(x + A) * eta(x^2 + A) * eta(x^3 + A) * eta(x^6 + A)), n))}; /* _Michael Somos_, Feb 17 2017 */
%o (Magma) A := Basis( ModularForms( Gamma0(6), 2), 59); A[1] + 6*A[2] + 6*A[3]; /* _Michael Somos_, Feb 17 2017 */
%Y Cf. A004011, A004016, A282544.
%K nonn
%O 0,2
%A _N. J. A. Sloane_, Jan 31 2007
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