A125510 Theta series of 4-dimensional lattice QQF.4.g.

1, 6, 6, 42, 6, 36, 42, 48, 6, 150, 36, 72, 42, 84, 48, 252, 6, 108, 150, 120, 36, 336, 72, 144, 42, 186, 84, 474, 48, 180, 252, 192, 6, 504, 108, 288, 150, 228, 120, 588, 36, 252, 336, 264, 72, 900, 144, 288, 42, 342, 186, 756, 84, 324, 474, 432, 48, 840, 180, 360, 252, 372

Offset

0,2

Comments

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

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

Links

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

David Broadhurst, Feynman integrals, L-series and Kloosterman moments, arXiv:1604.03057 [physics.gen-ph], 2016.

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

Formula

Expansion of a(x) * a(x^2) in powers of x where a() is a cubic AGM theta function. - Michael Somos, Feb 10 2011

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

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

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

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

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

G.f. A(x) = (F(x) + 3*F(x^3)) / 4 where F() = g.f. of A004011. - Michael Somos, Feb 17 2017

a(n) = A282544(2*n). - Michael Somos, Feb 18 2017

Sum_{k=1..n} a(k) ~ Pi^2 * n^2 / 3. - Vaclav Kotesovec, Dec 29 2023

Example

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 + ...
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 + ...

Mathematica

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 *)
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 *)

Prog

(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 */
(PARI) {et(n)=eta(q^n+O(q^(nt+1))); }
{nt=5000; et16=et(1)*et(6); et23=et(2)*et(3);
Eq105=(et16*et23)^2;
Eq135=(et23^3/et16)^3+q*(et16^3/et23)^3;
ans=Vec(Eq135/Eq105);
for(n=0, nt, print(n" "ans[n+1])); } /* David Broadhurst, Apr 12 2016 */
(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 */
(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 */
(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 */
(Magma) A := Basis( ModularForms( Gamma0(6), 2), 59); A[1] + 6*A[2] + 6*A[3]; /* Michael Somos, Feb 17 2017 */

Crossrefs

Cf. A004011, A004016, A282544.

Sequence in context: A279333 A278734 A279535 * A117859 A229159 A102901

Adjacent sequences: A125507 A125508 A125509 * A125511 A125512 A125513

Keyword

nonn

Author

N. J. A. Sloane, Jan 31 2007

Status

approved