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).

A282544 Expansion of (phi(x)^4 + 3*phi(x^3)^4) / 4 in powers of x where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, 2, 6, 14, 6, 12, 42, 16, 6, 50, 36, 24, 42, 28, 48, 84, 6, 36, 150, 40, 36, 112, 72, 48, 42, 62, 84, 158, 48, 60, 252, 64, 6, 168, 108, 96, 150, 76, 120, 196, 36, 84, 336, 88, 72, 300, 144, 96, 42, 114, 186, 252, 84, 108, 474, 144, 48, 280, 180, 120, 252
Offset: 0

Views

Author

Michael Somos, Feb 17 2017

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
a(n) is the number of solutions in integers to n = x^2 + y^2 + z^2 + w^2 where x + y + z = 3m is a multiple of 3. - Michael Somos, Jun 23 2018

Examples

			G.f. = 1 + 2*x + 6*x^2 + 14*x^3 + 6*x^4 + 12*x^5 + 42*x^6 + 16*x^7 + 6*x^8 + ...
a(4) = 6 with solutions (x, y, z, w) = {(1, 1, 1, 1), (1, 1, 1, -1), (0, 0, 0, 2)} and their negatives. - _Michael Somos_, Jun 23 2018

Links

Crossrefs

Cf. A013661, A033712, A125510.
Cf. A000122, A000700, A004016, A005882, A005928, A010054, A121373.

Programs

  • Magma
    A := Basis( ModularForms( Gamma0(12), 2), 60); A[1] + 2*A[2] + 6*A[3] + 14*A[4] + 6*A[5];
  • Mathematica
    a[ n_] := If[ n < 1, Boole[n == 0], 2 DivisorSum[n, # {1, 1, 2, 0, 1, 2, 1, 0, 2, 1, 1, 0}[[Mod[#, 12, 1]]] &]];
a[ n_] := If[ n < 1, Boole[n == 0], 2 Times @@ (Which[# < 3, 2 + (-1)^#, # == 3, 3^(#2 + 1) - 2, True, (#^(#2 + 1) - 1) / (# - 1)] & @@@ FactorInteger@n)];
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, x]^4 + 3 * EllipticTheta[ 3, 0, x^3]^4) / 4, {x, 0, n}];
  • PARI
    {a(n) = if( n<1, n==0, 2 * sumdiv(n, d, d * [0, 1, 1, 2, 0, 1, 2, 1, 0, 2, 1, 1][d%12+1]))};
  • PARI
    {a(n) = if( n<1, n==0, my(A = factor(n), p, e); 2 * prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 3, p==3, 3^(e+1) - 2, (p^(e+1) - 1) / (p - 1))))};
  • PARI
    {a(n) = if( n<0, 0, my(A); A = x * O(x^n); polcoeff( (sum(k=1, sqrtint(n), 2 * x^k^2, 1 + A)^4 + 3 * sum(k=1, sqrtint(n\3), 2 * x^(3*k^2), 1 + A)^4) / 4, n))};

Formula

Expansion of a(x^2) * phi(x) * phi(x^3) in powers of x where a() is a cubic AGM theta function and phi() is a Ramanujan theta function.
Expansion of (chi(x) * chi(x^3))^3 * (psi(x)^4 + 3*x*psi(x^3)^4) in powers of x where psi(), chi() are Ramanujan theta functions.
a(n) = 2*b(n) where b() is multiplicative with a(0) = 1, b(2^e) = 3 if e>0, b(3^e) = 3^(e+1) - 2, b(p^e) = (p^(e+1) - 1) / (p - 1) if p>3.
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 12 (t/i)^2 f(t) where q = exp(2 Pi i t).
G.f.: ((Sum_{k in Z} x^k^2)^4 + 3 * (Sum_{k in Z} x^(3*k^2))^4) / 4.
G.f.: 1 + 2 * Sum_{k>0} F(k, x) + 6 * Sum_{k>0} F(3*k, x) where F(k, x) = x^k / (1 + (-x)^k)^2.
G.f.: 1 + 2 * Sum_{k>0} F(k, x) + 2 * Sum_{k>0} F(3*k, x) where F(k, x) = k * x^k / (1 + (-x)^k).
a(2*n) = A125510(n). a(n) = A033712(2*n).
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/6 = 1.644934... (A013661). - Amiram Eldar, Dec 29 2023
Showing 1-2 of 2 results.

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