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A008430 Theta series of D_8 lattice. 2
1, 112, 1136, 3136, 9328, 14112, 31808, 38528, 74864, 84784, 143136, 149184, 261184, 246176, 390784, 395136, 599152, 550368, 859952, 768320, 1175328, 1078784, 1513152, 1362816, 2096192, 1764112, 2496928, 2289280, 3208832, 2731680, 4007808, 3336704 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..5000

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

N. 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.

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.

FORMULA

G.f.: (theta_3(q^(1/2))^8 + theta_4(q^(1/2))^8)/2.

a(n) = A000143(2n).

EXAMPLE

1 + 112*q^2 + 1136*q^4 + 3136*q^6 + 9328*q^8 + ...

MATHEMATICA

a[n_] := 16*DivisorSum[n, #^3*(8 - Mod[#, 2]) &]; a[0] = 1; Table[a[n], {n, 0, 40}] (* Jean-Fran├žois Alcover, Dec 02 2015, adapted from PARI *)

PROG

(PARI) {a(n)=if(n<1, n==0, 16*sumdiv(n, d, d^3*(8-d%2)))} /* 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))^8, n))} /* Michael Somos, Nov 03 2006 */

CROSSREFS

Cf. A000143, A008427 (dual), A109773.

Sequence in context: A206318 A206311 A235311 * A249004 A249469 A234673

Adjacent sequences:  A008427 A008428 A008429 * A008431 A008432 A008433

KEYWORD

nonn,easy,changed

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified December 16 07:41 EST 2017. Contains 296076 sequences.