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 A291124 #23 Feb 16 2025 08:33:50
%S A291124 1,8,16,-32,-144,-16,448,192,-912,-88,2016,-352,-4032,176,5504,64,
%T A291124 -7056,400,12112,352,-18144,-768,21312,-448,-25536,-968,35168,1216,
%U A291124 -49536,1584,56448,-1280,-56208,1408,78624,-384,-109008,-1296,109760,-704,-114912,-1584
%N A291124 Expansion of phi(x)^6 * phi(-x)^2 in powers of x where phi() is a Ramanujan theta function.
%C A291124 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700)
%H A291124 G. C. Greubel, <a href="/A291124/b291124.txt">Table of n, a(n) for n = 0..1000</a>
%H A291124 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A291124 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A291124 Expansion of (eta(q^2)^7 / (eta(q)^2 * eta(q^4)^3))^4 in powers of q.
%F A291124 Euler transform of period 4 sequence [8, -20, 8, -8, ...].
%F A291124 G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 512 (t/i)^4 g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A045820.
%F A291124 G.f.: Product_{k>0} (1 - x^(2*k))^28 / ((1 - x^k)^8 * (1 - x^(4*k))^12).
%F A291124 a(2*n + 1) = 8 * A030211(n). a(4*n + 2) = 16 * A045823(n).
%F A291124 a(2*n) = 16 * (-1)^n * (-sigma_3(n) + sigma_3(n/4)) where sigma_3(n) is the sum of the cubes of the divisors of n if n is an integer else 0.
%F A291124 Convolution square of A207541.
%e A291124 G.f. = 1 + 8*x + 16*x^2 - 32*x^3 - 144*x^4 - 16*x^5 + 448*x^6 + 192*x^7 + ...
%t A291124 a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x]^6 EllipticTheta[ 4, 0, x]^2, {x, 0, n}];
%t A291124 a[ n_] := SeriesCoefficient[ (QPochhammer[x^2]^7 / (QPochhammer[ x]^2 QPochhammer[ x^4]^3))^4, {x, 0, n}];
%o A291124 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^7 / (eta(x + A)^2 * eta(x^4 + A)^3))^4, n))};
%o A291124 (PARI) lista(nn) = {q='q+O('q^nn); Vec((eta(q^2)^7/(eta(q)^2*eta(q^4)^3))^4)} \\ _Altug Alkan_, Mar 21 2018
%o A291124 (Magma) A := Basis( ModularForms( Gamma0(16), 4), 42); A[1] + 8*A[2] + 16*A[3] - 32*A[4] - 144*A[5] - 16*A[6] + 448*A[7] + 192*A[8] - 912*A[9];
%Y A291124 Cf. A030211, A045820, A045823, A207541.
%K A291124 sign
%O A291124 0,2
%A A291124 _Michael Somos_, Aug 17 2017