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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.

A004011 Theta series of D_4 lattice; Fourier coefficients of Eisenstein series E_{gamma,2}.

Original entry on oeis.org

1, 24, 24, 96, 24, 144, 96, 192, 24, 312, 144, 288, 96, 336, 192, 576, 24, 432, 312, 480, 144, 768, 288, 576, 96, 744, 336, 960, 192, 720, 576, 768, 24, 1152, 432, 1152, 312, 912, 480, 1344, 144, 1008, 768, 1056, 288, 1872, 576, 1152, 96, 1368, 744, 1728, 336
Offset: 0

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Author

N. J. A. Sloane

Keywords

Comments

D_4 is also the Barnes-Wall lattice in 4 dimensions.
E_{gamma,2} is the unique normalized modular form for Gamma_0(2) of weight 2.
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
Ramanujan's Eisenstein series: P(q) (see A006352), Q(q) (A004009), R(q) (A013973).
Convolution square is A008658. - Michael Somos, Aug 21 2014
Expansion of 2*P(x^2) - P(x) in powers of x where P() is a Ramanujan Eisenstein series. - Michael Somos, Feb 16 2015
a(n) is the number of Hurwitz quaternions of norm n. - Michael Somos, Feb 16 2015

Examples

			G.f. = 1 + 24*x + 24*x^2 + 96*x^3 + 24*x^4 + 144*x^5 + 96*x^6 + 192*x^7 + 24*x^8 + ...
G.f. = 1 + 24*q^2 + 24*q^4 + 96*q^6 + 24*q^8 + 144*q^10 + 96*q^12 + 192*q^14 + 24*q^16 + ...

References

  • Bruce C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, 1998, see p. 148 Eq. (9.11).
  • Harvey Cohn, Advanced Number Theory, Dover Publications, Inc., 1980, p. 89. Eq. (1).
  • J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 119.
  • S. Ramanujan, Notebooks, Tata Institute of Fundamental Research, Bombay 1957 Vol. 1, see page 214.
  • N. J. A. Sloane, Seven Staggering Sequences, in Homage to a Pied Puzzler, E. Pegg Jr., A. H. Schoen and T. Rodgers (editors), A. K. Peters, Wellesley, MA, 2009, pp. 93-110.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A002388, A000118, A000593, A008658, A096727, A108092, A108096.
Cf. A004016, A005882, A005928.
Partial sums give A046949.
Cf. A108092 (convolution fourth root).

Programs

  • Magma
    Basis( ModularForms( Gamma0(2), 2), 54) [1]; /* Michael Somos, May 27 2014 */
  • Maple
    readlib(ifactors): with(numtheory): for n from 1 to 100 do if n mod 2 = 0 then m := n/ifactors(n)[2][1][1]^ifactors(n)[2][1][2] else m := n fi: printf(`%d,`,24*sigma(m)) od: # James Sellers, Dec 07 2000
  • Mathematica
    a[ n_] := If[ n < 0, 0, With[ {m = Floor @ Sqrt[4 n]}, SeriesCoefficient[ Sum[ q^( x^2 + y^2 + z^2 + t^2 + (x + y + z) t ), {x, -m, m}, {y, -m, m}, {z, -m, m}, {t, -m, m}] + O[q]^(n + 1), n]]]; (* Michael Somos, Jan 11 2011 *)
a[n_] := 24*Total[ Select[ Divisors[n], OddQ]]; a[0]=1; Table[a[n], {n, 0, 52}] (* Jean-François Alcover, Sep 12 2012 *)
a[ n_] := With[{m = InverseEllipticNomeQ @q}, SeriesCoefficient[ (1 + m) (EllipticK[ m] / (Pi/2))^2, {q, 0, n}]]; (* Michael Somos, Jun 04 2013 *)
a[ n_] := With[{m = InverseEllipticNomeQ @q}, SeriesCoefficient[ (1 - m/2) (EllipticK[ m] / (Pi/2))^2, {q, 0, 2 n}]]; (* Michael Somos, Jun 04 2013 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q]^4 + EllipticTheta[ 2, 0, q]^4, {q, 0, n}]; (* Michael Somos, Jun 04 2013 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q]^4 + EllipticTheta[ 4, 0, q]^4) / 2, {q, 0, 2 n}]; (* Michael Somos, Jun 04 2013 *)
  • PARI
    {a(n) = if( n<1, n==0, 24 * sumdiv( n, d, d%2 * d))}; /* Michael Somos, Apr 17 2000 */
  • PARI
    {a(n) = my(G); if( n<0, 0, G = [2, 1, 1, 1; 1, 2, 0, 0; 1, 0, 2, 0; 1, 0, 0, 2]; polcoeff( 1 + 2 * x * Ser(qfrep( G, n, 1)), n))}; /* Michael Somos, Sep 11 2007 */
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^20 / (eta(x + A) * eta(x^4 + A))^8 + 16 * x * eta(x^4 + A)^8 / eta(x^2 + A)^4, n))}; /* Michael Somos, Oct 21 2017 */
  • Python
    from sympy import divisors
def a(n): return 1 if n==0 else 24*sum(d for d in divisors(n) if d%2)
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 24 2017
  • Python
    from math import prod
from sympy import factorint
def A004011(n): return 24*prod((p**(e+1)-1)//(p-1) for p, e in factorint(n).items() if p > 2) if n else 1 # Chai Wah Wu, Nov 13 2022
  • Sage
    ModularForms( Gamma0(2), 2, prec=54).0; # Michael Somos, Jun 04 2013

Formula

a(0) = 1; if n>0 then a(n) = 24 (Sum_{d|n, d odd, d>0} d) = 24 * A000593(n).
G.f.: 1 + 24 Sum_{n>0} n x^n /(1 + x^n). a(n) = A000118(2*n) = A096727(2*n).
G.f.: (1/2) * (theta_3(z)^4 + theta_4(z)^4) = theta_3(2z)^4 + theta_2(2z)^4 = Sum_{k>=0} a(k) * x^(2*k).
G.f.: Sum_{a, b, c, d in Z} x^(a^2 + b^2 + c^2 + d^2 + a*d + b*d + c*d). - Michael Somos, Jan 11 2011
Expansion of (1 + k^2) * K(k^2)^2 / (Pi/2)^2 in powers of nome q. - Michael Somos, Jun 10 2006
Expansion of (1 - k^2/2) * K(k^2)^2 / (Pi/2)^2 in powers of nome q^2. - Michael Somos, Mar 14 2012
Expansion of b(x) * b(x^2) + 3 * c(x) * c(x^2) in powers of x where b(), c() are cubic AGM theta functions. - Michael Somos, Jan 11 2011
Expansion of b(x) * b(x^2) + c(x) * c(x^2) / 3 in powers of x^3 where b(), c() are cubic AGM theta functions. - Michael Somos, Mar 14 2012
G.f. is a period 1 Fourier series which satisfies f(-1 / (2 t)) = 2 (t/i)^2 f(t) where q = exp(2 Pi i t). - Michael Somos, Sep 11 2007
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u^2 - 2*u*v - 7*v^2 - 8*v*w + 16*w^2. - Michael Somos, May 29 2005
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u1^2 + 4*u2^2 + 9*u3^2 + 36*u6^2 - 2*u1*u2 - 10*u1*u3 + 10*u1*u6 + 10*u2*u3 - 40*u2*u6 - 18*u3*u6. - Michael Somos, Sep 11 2007
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2 = 9.869604... (A002388). - Amiram Eldar, Dec 29 2023

Extensions

Additional comments from Barry Brent (barryb(AT)primenet.com)

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