This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A008428 #30 Feb 16 2025 08:32:32
%S A008428 1,60,252,544,1020,1560,2080,3264,4092,4380,6552,8160,8224,10200,
%T A008428 12480,14144,16380,17400,18396,24480,26520,23040,31200,35904,32800,
%U A008428 39060,42840,44608,49344,50520,54080,65280,65532,57600,73080,84864,74460,82200,93600,92480
%N A008428 Theta series of D_6 lattice.
%C A008428 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%D A008428 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 118.
%H A008428 G. C. Greubel, <a href="/A008428/b008428.txt">Table of n, a(n) for n = 0..1000</a>
%H A008428 G. Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/D6.html">Home page for this lattice</a>
%H A008428 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A008428 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A008428 G.f.: (theta_3(q^(1/2))^6 + theta_4(q^(1/2))^6)/2
%F A008428 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
%F A008428 a(n) = A000141(2*n).
%F A008428 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
%e A008428 G.f. = 1 + 60*x + 252*x^2 + 544*x^3 + 1020*x^4 + 1560*x^5 + 2080*x^6 + ...
%e A008428 G.f. = 1 + 60*q^2 + 252*q^4 + 544*q^6 + 1020*q^8 + 1560*q^10 + 2080*q^12 + ...
%t A008428 a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x]^6, {x, 0, 2 n}]; (* _Michael Somos_, Aug 26 2015 *)
%o A008428 (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 */
%o A008428 (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 */
%o A008428 (Magma) A := Basis( ModularForms( Gamma1(8), 3), 80); A[1] + 60*A[3] + 252*A[5] + 544*A[7]; /* _Michael Somos_, Aug 26 2015 */
%Y A008428 Cf. A000141, A008425.
%K A008428 nonn,easy
%O A008428 0,2
%A A008428 _N. J. A. Sloane_