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A008428 Theta series of D_6 lattice. 3
1, 60, 252, 544, 1020, 1560, 2080, 3264, 4092, 4380, 6552, 8160, 8224, 10200, 12480, 14144, 16380, 17400, 18396, 24480, 26520, 23040, 31200, 35904, 32800, 39060, 42840, 44608, 49344, 50520, 54080, 65280, 65532, 57600, 73080, 84864, 74460, 82200, 93600, 92480 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

REFERENCES

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 118.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

G. Nebe and N. J. A. Sloane, Home page for this lattice

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

G.f.: (theta_3(q^(1/2))^6 + theta_4(q^(1/2))^6)/2

Expansion of ( phi(q)^6 + phi(-q)^6 ) / 2 in powers of q^2 where phi() is a Ramanujan theta function. - Michael Somos, Sep 14 2007

a(n) = A000141(2*n).

G.f. is a period 1 Fourier series that satisfies f(-1 / (8 t)) = 12 (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A008425. - Michael Somos, Aug 26 2015

EXAMPLE

G.f. = 1 + 60*x + 252*x^2 + 544*x^3 + 1020*x^4 + 1560*x^5 + 2080*x^6 + ...

G.f. = 1 + 60*q^2 + 252*q^4 + 544*q^6 + 1020*q^8 + 1560*q^10 + 2080*q^12 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x]^6, {x, 0, 2 n}]; (* Michael Somos, Aug 26 2015 *)

PROG

(PARI) {a(n) = if( n<1, n==0, 4 * sumdiv(n, d, d^2 * (16 * kronecker(-4, n/d) - kronecker(-4, d))))}; /* Michael Somos, Nov 03 2006 */

(PARI) {a(n) = if( n<0, 0, n*=2; polcoeff( sum(k=1, sqrtint(n), 2*x^k^2, 1 + x * O(x^n))^6, n))}; /* Michael Somos, Nov 03 2006 */

(MAGMA) A := Basis( ModularForms( Gamma1(8), 3), 80); A[1] + 60*A[3] + 252*A[5] + 544*A[7]; /* Michael Somos, Aug 26 2015 */

CROSSREFS

Cf. A000141, A008425.

Sequence in context: A019285 A261970 A206144 * A206232 A075295 A189542

Adjacent sequences:  A008425 A008426 A008427 * A008429 A008430 A008431

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified December 7 20:04 EST 2021. Contains 349585 sequences. (Running on oeis4.)