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 A004008 M5388 #29 Feb 16 2025 08:32:27
%S A004008 1,126,756,2072,4158,7560,11592,16704,24948,31878,39816,55944,66584,
%T A004008 76104,99792,116928,133182,160272,177660,205128,249480,265104,281736,
%U A004008 350784,382536,390726,470232,505568,532800,615384,640080,701568,799092,809424,853776
%N A004008 Expansion of theta series of E_7 lattice in powers of q^2.
%C A004008 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%D A004008 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 125. Equation (112)
%D A004008 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A004008 G. C. Greubel, <a href="/A004008/b004008.txt">Table of n, a(n) for n = 0..1000</a>
%H A004008 G. Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/E7.html">Home page for this lattice</a>
%H A004008 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A004008 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A004008 Expansion of phi(q)^3 * (phi(q)^4 + 7 * 16 * q * psi(q^2)^4) in powers of q where phi(), psi() are Ramanujan theta functions. - _Michael Somos_, Oct 24 2006
%F A004008 G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 2^(1/2) (t / i)^(7/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A003781. - _Michael Somos_, Aug 27 2013
%F A004008 Convolution of A005875 and A228746. - _Michael Somos_, Apr 21 2015
%e A004008 G.f. = 1 + 126*x + 756*x^2 + 2072*x^3 + 4158*x^4 + 7560*x^5 + 11592*x^6 + ...
%e A004008 G.f. = 1 + 126*q^2 + 756*q^4 + 2072*q^6 + 4158*q^8 + 7560*q^10 + 11592*q^12 + ...
%t A004008 a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q]^3 ( 8 EllipticTheta[ 3, 0, q]^4 - 7 EllipticTheta[ 4, 0, q]^4), {q, 0, n}]; (* _Michael Somos_, Aug 27 2013 *)
%t A004008 a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q]^3 ( EllipticTheta[ 3, 0, q]^4 + 7 EllipticTheta[ 2, 0, q]^4), {q, 0, n}]; (* _Michael Somos_, Apr 21 2015 *)
%o A004008 (PARI) {a(n) = my(A); if( n<0, 0, A = sum(k=1, sqrtint(n), 2 * x^k^2, 1 + x * O(x^n)); polcoeff( A^3 * (8 * A^4 - 7 * subst(A, x, -x)^4), n))}; /* _Michael Somos_, Oct 24 2006 */
%o A004008 (PARI) {a(n) = my(G); if( n<1, n==0, G = [2, -1, 0, 0, 0, 0, 0; -1, 2, -1, 0, 0, 0, 0; 0, -1, 2, -1, 0, 0, 0; 0, 0, -1, 2, -1, 0, -1; 0, 0, 0, -1, 2, -1, 0; 0, 0, 0, 0, -1, 2, 0; 0, 0, 0, -1, 0, 0, 2]; 2 * qfrep( G, n, 1)[n])}; /* _Michael Somos_, Jun 11 2007 */
%o A004008 (Magma) A := Basis( ModularForms( Gamma0(4), 7/2), 50); A[1] + 126*A[2]; /* _Michael Somos_, Jun 09 2014 */
%Y A004008 Cf. A003781, A005875, A228746.
%K A004008 nonn,easy
%O A004008 0,2
%A A004008 _N. J. A. Sloane_