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 A014972 #35 Feb 16 2025 08:32:33
%S A014972 1,16,128,704,3072,11488,38400,117632,335872,904784,2320128,5702208,
%T A014972 13504512,30952544,68901888,149403264,316342272,655445792,1331327616,
%U A014972 2655115712,5206288384,10049485312,19115905536,35867019904,66437873664
%N A014972 Expansion of (theta_3(q) / theta_4(q) )^4 in powers of q; also of 1 / (1 - lambda(z)).
%C A014972 The relation with A092877 is equivalent to eta(q^2)^24 = eta(q)^16 * eta(q^4)^8 + 16 * eta(q)^8 * eta(q^4)^16. - _Michael Somos_, Apr 11 2004
%D A014972 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 102.
%H A014972 Seiichi Manyama, <a href="/A014972/b014972.txt">Table of n, a(n) for n = 0..1000</a>
%H A014972 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EllipticLambdaFunction.html">Elliptic Lambda Function</a>
%F A014972 Expansion of 1 / (1 - lambda(t)) = 1 / lambda(-1 / t) in powers of q = exp(Pi i t).
%F A014972 Expansion of (phi(q) / phi(-q))^4 = (phi(-q^2) / phi(-q))^8 = (phi(q) / phi(-q^2))^8 = (f(q) / f(-q))^8 = (chi(q)/ chi(-q))^8 = (psi(q) / psi(-q))^8 in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions.
%F A014972 Expansion of (eta(q^2)^3 / (eta(q^4) * eta(q)^2))^8 in powers of q. - _Michael Somos_, Apr 11 2004
%F A014972 Euler transform of period 4 sequence [ 16, -8, 16, 0, ...]. - _Michael Somos_, Apr 11 2004
%F A014972 G.f. A(x) satisfies A(-x) = 1 / A(x). Also 0 = f(A(x), A(x^2)) where f(u, v) = (u - 1)^2 + 16 * u*v * (1 - v). - _Michael Somos_, Apr 11 2004
%F A014972 G.f.: (Product_{k>0} (1 + x^(2*k - 1)) / (1 - x^(2*k - 1)))^8 = exp( 16 * Sum_{k>0} x^(2*k - 1) * sigma(2*k - 1) / (2*k - 1)). - _Michael Somos_, Apr 11 2004
%F A014972 a(n) = 16 * A092877(n) unless n = 0. a(n) = A132136(n) unless n = 0. Convolution inverse of A128692.
%F A014972 Empirical : Sum_{n >=1} exp(-2*Pi)^(n-1)*(-1)^(n+1)*a(n) = -16+12*2^(1/2). - _Simon Plouffe_, Feb 20 2011
%F A014972 a(n) ~ exp(2*Pi*sqrt(n)) / (32 * n^(3/4)). - _Vaclav Kotesovec_, Aug 28 2015
%e A014972 G.f. = 1 + 16*q + 128*q^2 + 704*q^3 + 3072*q^4 + 11488*q^5 + 38400*q^6 + 117632*q^7 + ...
%t A014972 a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] / EllipticTheta[ 4, 0, q])^4, {q, 0, n}]; (* _Michael Somos_, Jun 25 2014 *)
%t A014972 a[ n_] := SeriesCoefficient[ (QPochhammer[ q^2]^3 / (QPochhammer[ q^4] QPochhammer[ q]^2))^8, {q, 0, n}]; (* _Michael Somos_, Jun 25 2014 *)
%t A014972 nmax=60; CoefficientList[Series[Product[((1+x^(2*k+1))/(1-x^(2*k+1)))^8,{k,0,nmax}],{x,0,nmax}],x] (* _Vaclav Kotesovec_, Aug 28 2015 *)
%o A014972 (PARI) {a(n) = if( n<0, 0, polcoeff( exp( 16 * sum( k=1, (n+1)\2, sigma(2*k - 1) / (2*k - 1) * x^(2*k - 1), x * O(x^n))), n))}; /* _Michael Somos_, Apr 11 2004 */
%o A014972 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^3 / (eta(x + A)^2 * eta(x^4 + A)))^8, n))}; /* _Michael Somos_, Apr 11 2004 */
%Y A014972 Cf. A005798, A007248, A029845, A092877, A128692, A132136.
%K A014972 nonn
%O A014972 0,2
%A A014972 _N. J. A. Sloane_