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 A258100 #10 Feb 16 2025 08:33:25
%S A258100 1,1,0,-1,-2,0,4,5,0,-10,-12,0,20,26,0,-39,-50,0,76,92,0,-140,-168,0,
%T A258100 244,295,0,-415,-496,0,696,818,0,-1140,-1332,0,1820,2126,0,-2861,
%U A258100 -3324,0,4448,5126,0,-6816,-7824,0,10292,11793,0,-15372,-17548,0,22756
%N A258100 Expansion of c(q) * c(q^3) / c(q^2)^2 in powers of q where c() is a cubic AGM theta function.
%C A258100 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A258100 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%H A258100 G. C. Greubel, <a href="/A258100/b258100.txt">Table of n, a(n) for n = 0..1000</a>
%H A258100 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A258100 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A258100 Expansion of (psi(q) * f(-q^9)^3) / (chi(-q^3)^2 * psi(q^3)^4) in powers of q where psi(), chi(), f() are Ramanujan theta functions.
%F A258100 Expansion of eta(q^2)^2 * eta(q^3)^2 * eta(q^9)^3 / (eta(q) * eta(q^6)^6) in powers of q.
%F A258100 Euler transform of period 18 sequence [ 1, -1, -1, -1, 1, 3, 1, -1, -4, -1, 1, 3, 1, -1, -1, -1, 1, 0, ...].
%F A258100 a(n) = (-1)^n * A164616(n). a(3*n) = A128641(n). a(3*n + 1) = A258099(n). a(3*n + 2) = 0.
%F A258100 Convolution invserse is A182034.
%e A258100 G.f. = 1 + q - q^3 - 2*q^4 + 4*q^6 + 5*q^7 - 10*q^9 - 12*q^10 + 20*q^12 + ...
%t A258100 a[ n_] := SeriesCoefficient[ QPochhammer[ q^9]^3 EllipticTheta[ 2, 0, q^(1/2)] QPochhammer[ q^3]^2 / (2 q^(1/8) QPochhammer[ q^6]^6), {q, 0, n}];
%t A258100 a[ n_] := SeriesCoefficient[ 4 q QPochhammer[ q^9]^3 EllipticTheta[ 2, 0, q^(1/2)] / (QPochhammer[ q^3] EllipticTheta[ 2, 0, q^(3/2)]^3), {q, 0, n}];
%o A258100 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^3 + A)^2 * eta(x^9 + A)^3 / (eta(x + A) * eta(x^6 + A)^6), n))};
%Y A258100 Cf. A128641, A164616, A182034, A258099.
%K A258100 sign
%O A258100 0,5
%A A258100 _Michael Somos_, May 20 2015