A004011 Theta series of D_4 lattice; Fourier coefficients of Eisenstein series E_{gamma,2}. (Formerly M5140)

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

0,2

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

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

T. D. Noe, Table of n, a(n) for n = 0..10000

Megumi Asada, Bruce Fang, Eva Fourakis, Sarah Manski, Nathan McNew, Steven J. Miller, Gwyneth Moreland, Ajmain Yamin, and Sindy Xin Zhang, Avoiding 3-Term Geometric Progressions in Hurwitz Quaternions, Williams College (2023).

Barry Brent, Quadratic Minima and Modular Forms, Experimental Mathematics, Vol. 7, No. 3 (1998), 257-274.

Michael Gilleland, Some Self-Similar Integer Sequences.

Nadia Heninger, E. M. Rains, and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, arXiv:math/0509316 [math.NT], 2005-2006; J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.

Masao Koike, Modular forms on non-compact arithmetic triangle groups, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021. I wrote 2005 on the first page but the internal evidence suggests 1997.]

Gabriele Nebe and N. J. A. Sloane, Home page for D_4 lattice

N. J. A. Sloane, The 24 minimal vectors form the 24-cell polytope.

N. J. A. Sloane, Seven Staggering Sequences.

Eric Weisstein's World of Mathematics, 24-Cell.

Eric Weisstein's World of Mathematics, Barnes-Wall Lattice.

Eric Weisstein's World of Mathematics, Eisenstein Series.

Wikipedia, Hurwitz quaternion.

Index entries for "core" sequences.

Index entries for sequences related to Barnes-Wall lattices.

Index entries for sequences related to D_4 lattice.

Index entries for sequences related to Eisenstein series.

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

Example

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 + ...

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 A. 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 *)

Prog

(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 */
(Sage) ModularForms( Gamma0(2), 2, prec=54).0; # Michael Somos, Jun 04 2013
(Magma) Basis( ModularForms( Gamma0(2), 2), 54) [1]; /* Michael Somos, May 27 2014 */
(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

Crossrefs

Cf. A002388, A000118, A000593, A008658, A096727, A108092, A108096.

Cf. A004016, A005882, A005928.

Partial sums give A046949.

Cf. A108092 (convolution fourth root).

Sequence in context: A022358 A122505 A103640 * A334570 A056465 A056455

Adjacent sequences: A004008 A004009 A004010 * A004012 A004013 A004014

Keyword

nonn,easy,core,nice

Author

N. J. A. Sloane

Extensions

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

Status

approved