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 A062248 #20 Apr 20 2025 04:23:58
%S A062248 1,-48,-216,1536,-1560,-3024,13824,-8736,-14040,41712,-27216,-31968,
%T A062248 112128,-51072,-74304,193536,-113880,-117936,375408,-165984,-220752,
%U A062248 528384,-287712,-292032,898560,-375024,-474768,1126464,-598848,-585360,1741824,-722400,-898776
%N A062248 Expansion of a Schwarzian ({f_{27|3}, tau} / (4*Pi)^2) in powers of q^3.
%C A062248 The q-series f_{27|3} is the g.f. for A062246. This is given on page 274 of McKay and Sebbar along with equation (8.2) which gives an expression for the g.f. A(q) of this sequence, but the left side is A(q^3) and the right side is A(q). - _Michael Somos_, Aug 12 2014
%C A062248 Ramanujan theta function: f(-q) (see A010815). Ramanujan Lambert series: Q(q) = E_4(q) (see A004009).
%C A062248 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%H A062248 G. C. Greubel, <a href="/A062248/b062248.txt">Table of n, a(n) for n = 0..1000</a>
%H A062248 J. McKay and A. Sebbar, <a href="http://dx.doi.org/10.1007/s002080000116">Fuchsian groups, automorphic functions and Schwarzians</a>, Math. Ann., 318 (2000), 255-275.
%F A062248 Expansion of Q(q^3) - 48 * q * f(-q^3)^8 - 216 * q^2 * (f(-q) * f(-q)^9)^6 / f(-q^3)^4 in powers of q where Q(), f() are Ramanujan q-series. - _Michael Somos_, Aug 12 2014
%F A062248 Expansion of (a(q)^4 - 18 * a(q)^3*a(q^3) + 60 * a(q)^2*a(q^3)^2 - 54 * a(q)*a(q^3)^3 + 9 * a(q^3)^4) / -2 where a() is a cubic AGM theta function. - _Michael Somos_, Aug 12 2014
%F A062248 Expansion of b(q)^4 - 12 * b(q)^3*c(q^3) - 66 * b(q)^2*c(q^3)^2 - 36 * b(q)*c(q^3)^3 + 9 * c(q^3)^4 in powers of q where b(), c() are cubic AGM theta functions. - _Michael Somos_, Aug 12 2014
%F A062248 Expansion of E_4(q^3) - 48 * eta(q^3)^8 - 216 * eta(q)^6 * eta(q^9)^6 / eta(q^3)^4 in powers of q. [McKay and Sebbar, equation (8.2)] - _Michael Somos_, Aug 12 2014
%F A062248 G.f. is a period 1 Fourier series which satisfies f(-1 / (9 t)) = 81 (t/i)^4 f(t) where q = exp(2 Pi i t).
%F A062248 a(3*n) = A004009(n) -216 * A242042(3*n). a(3*n + 1) = -48 * A000731(n) -216 * A242042(3*n + 1). a(3*n + 2) = -216 * A242042(3*n + 2). - _Michael Somos_, Aug 12 2014
%e A062248 G.f. = 1 - 48*x - 216*x^2 + 1536*x^3 - 1560*x^4 - 3024*x^5 + 13824*x^6 + ...
%e A062248 G.f. = 1 - 48*q^3 - 216*q^6 + 1536*q^9 - 1560*q^12 - 3024*q^15 + 13824*q^18 + ...
%t A062248 QP = QPochhammer; A = x*O[x]^40; A1 = QP[x + A]^3; A3 = QP[x^3 + A]^4; A9 = x*QP[x^9 + A]^3; s = ((A1 + 3*A9)*(A1 + 9*A9)*(A1^2 + 27*A9^2) - 48*x*A3^3 - 216*(A1*A9)^2)/A3; CoefficientList[s, x] (* _Jean-François Alcover_, Nov 14 2015, adapted from _Michael Somos_'s PARI script *)
%t A062248 eta[q_] := q^(1/24)*QPochhammer[q]; E4[q] := 1; E4[q_] := 1 + 240 *Sum[k^3* q^k/(1 - q^k), {k, 1, 500}]; CoefficientList[Series[E4[q^3] - 48*eta[q^3]^8 - 216*(eta[q]*eta[q^9])^6/eta[q^3]^4, {q, 0, 50}], q] (* _G. C. Greubel_, May 01 2018 *)
%o A062248 (PARI) {a(n) = local(A, A1, A3, A9); if( n<0, 0, A = x * O(x^n); A1 = eta(x + A)^3; A3 = eta(x^3 + A)^4; A9 = x * eta(x^9 + A)^3; polcoeff( ((A1 + 3*A9) * (A1 + 9*A9) * (A1^2 + 27*A9^2) - 48*x*A3^3 - 216*(A1*A9)^2) / A3, n))}; /* _Michael Somos_, Aug 12 2014 */
%o A062248 (Magma) A := Basis( ModularForms( Gamma0(9), 8/2), 30); A[1] - 48*A[2] - 216*A[3] + 1536*A[4] - 1560*A[5]; /* _Michael Somos_, Aug 12 2014 */
%Y A062248 Cf. A000731, A004009, A062246, A242042.
%K A062248 sign
%O A062248 0,2
%A A062248 _N. J. A. Sloane_, Jul 01 2001
%E A062248 More terms from John McKay (mckay(AT)cs.concordia.ca), Apr 18 2004
%E A062248 More terms from _Michael Somos_, Aug 12 2014