cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A033712 theta3(z) * theta3(2*z) * theta3(3*z) * theta3(6*z).

Original entry on oeis.org

1, 2, 2, 6, 6, 4, 14, 8, 6, 26, 12, 16, 42, 12, 16, 44, 6, 20, 50, 16, 36, 56, 24, 16, 42, 30, 28, 78, 48, 36, 84, 40, 6, 80, 36, 48, 150, 44, 40, 100, 36, 36, 112, 48, 72, 148, 48, 48, 42, 50, 62, 124, 84, 52, 158, 64, 48, 144, 60, 64, 252, 60, 64, 200, 6, 88, 168, 64, 108
Offset: 0

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Author

N. J. A. Sloane

Keywords

Comments

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

Examples

			G.f. = 1 + 2*q + 2*q^2 + 6*q^3 + 6*q^4 + 4*q^5 + 14*q^6 + 8*q^7 + 6*q^8 + ...

References

  • J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 102, eq. 9.
  • L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 3, p. 225.

Links

Crossrefs

Cf. A125510, A282544.

Programs

  • Magma
    A := Basis( ModularForms( Gamma0(24), 2), 69); A[1] + 2*A[2] + 2*A[3] + 6*A[4] + 6*A[5] + 4*A[6] + 14*A[7] + 6*A[8]; /* Michael Somos, Apr 19 2015 */
  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^2] EllipticTheta[ 3, 0, q^3] EllipticTheta[ 3, 0, q^6], {q, 0, n}]; (* Michael Somos, Apr 19 2015 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = sum( k=1, sqrtint(n), 2 * x^k^2, 1 + x * O(x^n)); polcoeff( A * subst( A + x * O(x^(n\2)), x ,x^2) * subst( A + x * O(x^(n\3)), x, x^3) * subst( A + x * O(x^(n\6)), x, x^6), n))}; /* Michael Somos, May 30 2005 */
  • PARI
    {a(n) = my(G); if( n<0, 0, G = [1, 0, 0, 0; 0, 2, 0, 0; 0, 0, 3, 0; 0, 0, 0, 6 ]; polcoeff( 1 + 2 * x * Ser( qfrep( G, n)), n))}; /* Michael Somos, Apr 19 2015 */

Formula

Number of solutions to a^2 + 2*b^2 + 3*c^2 + 6*d^2 = n in integers.
Expansion of phi(q) * phi(q^2) * phi(q^3) * phi(q^6) in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Apr 19 2015
Expansion of (eta(q^2) * eta(q^4) * eta(q^6) * eta(q^12))^3 / (eta(q) * eta(q^3) * eta(q^8) * eta(q^24))^2 in powers of q.
Euler transform of period 24 sequence [2, -1, 4, -4, 2, -2, 2, -2, 4, -1, 2, -8, 2, -1, 4, -2, 2, -2, 2, -4, 4, -1, 2, -4, ...]. - Michael Somos, May 30 2005
G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = 24 (t/i)^2 f(t) where q = exp(2 Pi i t). - Michael Somos, Apr 19 2015
a(2*n) = A282544(n). a(4*n) = A125510(n).

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

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane, Jan 31 2007

Keywords

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).

Examples

			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 + ...

Links

Crossrefs

Cf. A004011, A004016, A282544.

Programs

  • Magma
    A := Basis( ModularForms( Gamma0(6), 2), 59); A[1] + 6*A[2] + 6*A[3]; /* Michael Somos, Feb 17 2017 */
  • 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 *)
  • 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 */

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
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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