login
This site is supported by donations to The OEIS Foundation.

 

Logo

110 people attended OEIS-50 (videos, suggestions); annual fundraising drive to start soon (donate); editors, please edit! (stack is over 300), your editing is more valuable than any donation.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A004011 Theta series of D_4 lattice; Fourier coefficients of Eisenstein series E_{gamma,2}.
(Formerly M5140)
18
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 (list; graph; refs; listen; history; text; internal format)
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).

REFERENCES

B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, see p. 148

Eq. (9.11).

H. 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.

Masao Koike, Modular forms on non-compact arithmetic triangle groups, preprint.

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

B. Brent, Quadratic Minima and Modular Forms, Experimental Mathematics, v.7 no.3, 257-274.

Michael Gilleland, Some Self-Similar Integer Sequences

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.

G. 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, Eisenstein Series

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

Index entries for "core" sequences

Index entries for sequences related to D_4 lattice

Index entries for sequences related to Eisenstein series

Index entries for sequences related to Barnes-Wall lattices

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

Convolution square is A008658. - Michael Somos, Aug 21 2014

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-Fran├žois 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) = if( n<1, n==0, qfrep([ 2, 1, 1, 1; 1, 2, 0, 0; 1, 0, 2, 0; 1, 0, 0, 2], n, 1)[n] * 2 )}; /* Michael Somos, Sep 11 2007 */

(Sage) ModularForms( Gamma0(2), 2, prec=33).0 ; # Michael Somos, Jun 04 2013

(MAGMA) Basis( ModularForms( Gamma0(2), 2), 33) [1] ; /* Michael Somos, May 27 2014 */

CROSSREFS

Cf. A000118, A000593, A096727, A108092, A108096. Partial sums give A046949.

Cf. A108092 (convolution fourth root).

Sequence in context: A022358 A122505 A103640 * A056465 A056455 A128378

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified October 31 17:19 EDT 2014. Contains 248868 sequences.