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A125510 Theta series of 4-dimensional lattice QQF.4.g. 3
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 (list; graph; refs; listen; history; text; internal format)
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

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-Fran├žois 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

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Last modified January 19 03:54 EST 2020. Contains 331031 sequences. (Running on oeis4.)