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A092848 Expansion of reciprocal of Hauptmodul for Gamma_0(18).

%I A092848 #26 Feb 16 2025 08:32:53
%S A092848 1,-1,0,2,-2,-1,4,-4,-1,8,-8,-2,14,-14,-4,24,-23,-6,40,-38,-10,63,-60,
%T A092848 -16,98,-92,-24,150,-140,-36,224,-208,-54,329,-304,-78,478,-440,-112,
%U A092848 684,-627,-160,968,-884,-224,1358,-1236,-312,1884,-1710,-432,2592,-2346,-590,3540,-3196,-801,4796,-4320,-1082,6454
%N A092848 Expansion of reciprocal of Hauptmodul for Gamma_0(18).
%C A092848 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A092848 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%D A092848 B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 345 Entry 1(i).
%H A092848 Seiichi Manyama, <a href="/A092848/b092848.txt">Table of n, a(n) for n = 0..10000</a>
%H A092848 S. Cooper, <a href="http://dx.doi.org/10.1007/s11139-011-9357-3">Sporadic sequences, modular forms and new series for 1/pi</a>, Ramanujan J. (2012).
%H A092848 W. Duke, <a href="http://dx.doi.org/10.1090/S0273-0979-05-01047-5">Continued fractions and modular functions</a>, Bull. Amer. Math. Soc. 42 (2005), 137-162. See page 155 Eq. (9.13)
%H A092848 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A092848 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A092848 Expansion of chi(-q) / chi(-q^3)^3 where chi() is a Ramanujan theta function.
%F A092848 Expansion of q^(-1/3) * c(q^2) / c(q) where c() is a cubic AGM theta function. - _Michael Somos_, Oct 04 2006
%F A092848 Expansion of q^(-1/3) * eta(q) * eta(q^6)^3 / (eta(q^2) * eta(q^3)^3) in powers of q.
%F A092848 Euler transform of period 6 sequence [-1, 0, 2, 0, -1, 0, ...].
%F A092848 Given g.f. A(x), then B(q) = q * A(q^3) satisfies 0 = f(B(q), B(q^2)) where f(u, v) = u^2 - v + 2*u*v^2.
%F A092848 Given g.f. A(x), then B(q) = q * A(q^3) satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (v^3 - v^2 + v) - u^3 * (1 + 2*v + 4*v^2).
%F A092848 G.f. is a period 1 Fourier series which satisfies f(-1 / (18 t)) = (1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A141094. - _Michael Somos_, Dec 07 2013
%F A092848 G.f.: Product_{k>0} (1 - x^(2*k - 1)) / (1 - x^(6*k - 3))^3.
%F A092848 G.f.: 1 / (1 + (x + x^2) / (1 + (x^2 + x^4) / (1 + (x^3 + x^6) / ...))).
%F A092848 a(n) = A062242(2*n + 1) = (-1)^n * A128111(n). Convolution inverse of A062242.
%F A092848 a(2*n + 1) = - A216046(n). Convolution square is A216046. - _Michael Somos_, Dec 07 2013
%F A092848 G.f.: T(0), where T(k) = 1 - (x^(k+1)+x^(2*k+2))/((x^(k+1)+x^(2*k+2))+1/T(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Oct 14 2013
%e A092848 G.f. = 1 - x + 2*x^3 - 2*x^4 - x^5 + 4*x^6 - 4*x^7 - x^8 + 8*x^9 + ...
%e A092848 G.f. = q - q^4 + 2*q^10 - 2*q^13 - q^16 + 4*q^19 - 4*q^22 - q^25 + 8*q^28 + ...
%t A092848 a[ n_] := SeriesCoefficient[ Product[ 1 - x^k, {k, 1, n, 2}] / Product[ 1 - x^k, {k, 3, n, 6}]^3, {x, 0, n}]; (* _Michael Somos_, Dec 07 2013 *)
%t A092848 a[ n_] := SeriesCoefficient[  QPochhammer[ x, x^2] / QPochhammer[ x^3, x^6]^3, {x, 0, n}]; (* _Michael Somos_, Dec 07 2013 *)
%o A092848 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^6 + A)^3 / (eta(x^2 + A) * eta(x^3 + A)^3), n))}; /* _Michael Somos_, Dec 07 2013 */
%o A092848 (PARI) {a(n) = my(A, m); if( n<0, 0, A = 1 + O(x); m=1; while( m<=n, m*=2; A = subst(A, x, x^2); A = sqrt(A + (x*A^2)^2) - x*A^2); polcoeff(A, n))};
%o A092848 (PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=0, (n-1)\2, (1 - x^(2*k + 1))^if(k%3==1, -2, 1), 1 + x * O(x^n)), n))};
%Y A092848 Cf. A062242, A128111, A141094, A216046.
%K A092848 sign
%O A092848 0,4
%A A092848 _Michael Somos_, Mar 07 2004