A108096 -id:A108096 - cp's OEIS frontend

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

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

N. J. A. Sloane

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

			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 + ...

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).

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.
Fern Gossow, Lyndon-like cyclic sieving and Gauss congruence, arXiv:2410.05678 [math.CO], 2024. See p. 26.
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.

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

Basis( ModularForms( Gamma0(2), 2), 54) [1]; /* Michael Somos, May 27 2014 */

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

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 *)

{a(n) = if( n<1, n==0, 24 * sumdiv( n, d, d%2 * d))}; /* Michael Somos, Apr 17 2000 */

{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 */

{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 */

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

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

ModularForms( Gamma0(2), 2, prec=54).0; # Michael Somos, Jun 04 2013

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

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

A008658 Theta series of direct sum of 2 copies of D_4 lattice in powers of q^2.

Original entry on oeis.org

1, 48, 624, 1344, 5232, 6048, 17472, 16512, 42096, 36336, 78624, 63936, 146496, 105504, 214656, 169344, 337008, 235872, 472368, 329280, 659232, 462336, 831168, 584064, 1178688, 756048, 1371552, 981120, 1799808, 1170720, 2201472

Offset: 0

N. J. A. Sloane

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

			G.f. = 1 + 48*x + 624*x^2 + 1344*x^3 + 5232*x^4 + 6048*x^5 + 17472*x^6 + ...
G.f. = 1 + 48*q^2+ 624*q^4 + 1344*q^6 + 5232*q^8 + 6048*q^10 + 17472*q^12 + ...

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 119.
Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 116, equ. (3) and p. 119, 10th equ.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
B. Brent, Quadratic Minima and Modular Forms, Experimental Mathematics, v.7 no.3 (1998), 257-274.
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.]
J. McKay and A. Sebbar, Fuchsian groups, automorphic functions and Schwarzians, Math. Ann., 318 (2000), 255-275.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Index entries for sequences related to D_4 lattice

Cf. A004009, A004011, A108096.

A := Basis( ModularForms( Gamma0(8), 4), 62); A[1] + 48*A[3] + 624*A[5]; /* Michael Somos, Aug 20 2014 */

a[ n_] := If[ n < 1, Boole[n == 0], 48 (DivisorSigma[3, n] + If[OddQ[n], 0, 4 DivisorSigma[3, n/2]])]; (* Michael Somos, Feb 19 2017 *)

{a(n) = if( n<1, n==0, 48 * (sigma(n, 3) + if( n%2, 0, 4*sigma(n/2, 3))))}; /* Michael Somos, Jul 16 2004 */

Fourier coefficients of E_{gamma,2}^2.
Convolution square of A004011. Convolution fourth power of A108096. - Michael Somos, Aug 20 2014
G.f.: (E_4(x) + 4*E_4(x^2)) / 5 where E_4() is the g.f. of A004009. [Ramanujan]. - Michael Somos, Feb 19 2017
Expansion of(2*phi(x)^4 - phi(-x)^4)^2 in powers of x where phi() is a Ramanujan theta function. - Michael Somos, Feb 19 2017
Expansion of phi(-x)^8 + 64*x * psi(x)^8 in powers of x where phi(), psi() are Ramanujan theta functions. - Michael Somos, Feb 19 2017
Expansion of (phi(-x)^4 + 8*x * psi(x^2)^4)^2 in powers of x^2 where phi(), psi() are Ramanujan theta functions. - Michael Somos, Feb 19 2017
a(n) = 48*b(n) where b() is multiplicative with b(2^e) = 1 + 12*(8^e - 1) / 7, b(p^e) = (p^(3*(e+1)) - 1) / (p^3 - 1) if p>2. - Michael Somos, Feb 19 2017

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

A108092 Coefficients of series whose 4th power is the theta series of D_4 (see A004011).

Original entry on oeis.org

1, 6, -48, 672, -10686, 185472, -3398304, 64606080, -1261584768, 25141699590, -509112525600, 10443131883360, -216500232587520, 4528450460408448, -95438941858567104, 2024550297637849728, -43190698219545864702, 925997705081213764608, -19940633776083900614736, 431091393800371703940576

Offset: 0

N. J. A. Sloane and Michael Somos, Jun 06 2005

sign

			More precisely, the theta series of D_4 begins 1 + 24*q^2 + 24*q^4 + 96*q^6 + 24*q^8 + 144*q^10 + 96*q^12 + ... and its 4th root is 1 + 6*q^2 - 48*q^4 + 672*q^6 - 10686*q^8 + 185472*q^10 - 3398304*q^12 + ...

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.

Vaclav Kotesovec, Table of n, a(n) for n = 0..730
N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
N. J. A. Sloane, Seven Staggering Sequences.
N. J. A. Sloane, Old and New Problems from 55 Years of the OEIS, Slides of talk in Doron Zeilberger's Experimental Math Seminar at Rutgers University, October 10 2019.

Cf. A004011, A108096.

CoefficientList[Series[(EllipticTheta[3,0,x]^4 + EllipticTheta[2,0,x]^4)^(1/4), {x, 0, 20}], x] (* Vaclav Kotesovec, Dec 10 2017 *)

a(n) ~ -(-1)^n * Gamma(1/4)^3 * exp(Pi*n) / (2^(15/4) * Pi^(5/2) * n^(5/4)). - Vaclav Kotesovec, Dec 10 2017

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

PARI

Python

Python

Sage

Formula

Extensions

A008658 Theta series of direct sum of 2 copies of D_4 lattice in powers of q^2.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A108092 Coefficients of series whose 4th power is the theta series of D_4 (see A004011).

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula