0,2

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.

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 119.

H. Cohn, Advanced Number Theory, Dover Publications, Inc., 1980, p. 89. Eq. (1).

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

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.

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

Michael Gilleland, Some Self-Similar Integer Sequences

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, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

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

a(0)=1; if n>0 then a(n)=24 (sum_{d|n, d odd, d>0} d).

G.f.: 1+24 Sum_{n>0} nx^n/(1+x^n).

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

Expansion of (1+k^2)K(k^2)^2/(pi/2)^2 in powers of nome q. - Michael Somos Jun 10 2006

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^(2k).

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^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

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 + ...

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: # from James A. Sellers Dec 07 2000

(PARI) a(n)=if(n<1, n==0, 24*sumdiv(n, d, d%2*d))

(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 */

a(n)=24*A000593(n), n>0. Partial sums give A046949. Cf. A108092, A108096.

A000118(2n)=A096727(2n)=a(n).

Cf. A108092 (fourth root).

Sequence in context: A022358 A122505 A103640 this_sequence A056465 A056455 A128378

Adjacent sequences: A004008 A004009 A004010 this_sequence A004012 A004013 A004014

nonn,easy,core,nice

njas

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