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 A008444 #43 Feb 16 2025 08:32:32
%S A008444 1,20,30,60,60,120,40,180,150,140,130,240,180,360,120,260,220,480,210,
%T A008444 400,360,240,360,660,200,620,240,600,540,600,240,640,630,720,320,780,
%U A008444 420,1080,600,480,650,840,360,1260,720,840,440,1380,660,860,630,640,1080,1560,400
%N A008444 Theta series of A_4 lattice.
%C A008444 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%D A008444 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 110.
%H A008444 John Cannon, <a href="/A008444/b008444.txt">Table of n, a(n) for n = 0..5000</a>
%H A008444 G. Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/A4.html">Home page for this lattice</a>
%H A008444 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A008444 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A008444 Expansion of f(-x)^5 / f(-x^5) + 25 * x * f(-x^5)^5 / f(-x) in powers of x where f() is a Ramanujan theta function. - _Michael Somos_, Feb 06 2011
%F A008444 Expansion of (1 / Pi) integral_{0 .. Pi/2} theta_3(z, q)^5 + theta_4(z, q)^5 dz in powers of q^2. - _Michael Somos_, Jan 01 2012
%F A008444 Coefficient of x^0 in the expansion f(x * q, q / x)^5 in powers of q^2 where f() is a Ramanujan theta function. - _Michael Somos_, Jan 01 2012
%F A008444 G.f. is a period 1 Fourier series which satisfies f(-1 / (5 t)) = 5^(3/2) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A023916. - _Michael Somos_, Feb 06 2011
%F A008444 A023916(5*n) = a(n) for all n in Z.
%e A008444 G.f. = 1 + 20*x + 30*x^2 + 60*x^3 + 60*x^4 + 120*x^5 + 40*x^6 + 180*x^7 + ...
%e A008444 G.f. = 1 + 20*q^2 + 30*q^4 + 60*q^6 + 60*q^8 + 120*q^10 + 40*q^12 + 180*q^14 + 150*q^16 + 140*q^18 + 130*q^20 + 240*q^22 + 180*q^24 + 360*q^26 + 120*q^28 + 260*q^30 + 220*q^32 + 480*q^34 + 210*q^36 + 400*q^38 + 360*q^40 + 240*q^42 + 360*q^44 + 660*q^46 + 200*q^48 + 620*q^50 + ...
%t A008444 a[ n_] := With[ {u1 = QPochhammer[ x], u5 = QPochhammer[ x^5]}, SeriesCoefficient[ u1^5/u5 + 25 x u5^5/u1, {x, 0, n}]]; (* _Michael Somos_, Nov 13 2014 *)
%t A008444 terms = 55; f[q_] = LatticeData["A4", "ThetaSeriesFunction"][-I Log[q]/Pi]; s = Series[f[q], {q, 0, 2 terms}]; CoefficientList[s, q^2][[1 ;; terms]] (* _Jean-François Alcover_, Jul 04 2017 *)
%o A008444 (Magma) L := Lattice("A", 4); A<q> := ThetaSeries(L, 120); A;
%o A008444 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^5 / eta(x^5 + A) + 25 * x * eta(x^5 + A)^5 / eta(x + A), n))}; /* _Michael Somos_, Feb 06 2011 */
%o A008444 (Magma) A := Basis( ModularForms( Gamma1(5), 2), 55) ; A[1] + 20*A[2] + 30*A[3]; /* _Michael Somos_, Nov 13 2014 */
%Y A008444 Cf. A000007, A000122, A004016, A004015, A008445, A008446, A008447, A008448, A008449 (Theta series of lattices A_0, A_1, A_2, A_3, A_5, ...).
%K A008444 nonn,nice
%O A008444 0,2
%A A008444 _N. J. A. Sloane_