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A114855 Expansion of q^(-1/3) * (eta(q) * eta(q^4))^2 / eta(q^2) in powers of q.

%I A114855 #24 Feb 16 2025 08:33:00
%S A114855 1,-2,0,0,0,4,0,0,-5,0,0,0,0,0,0,0,7,0,0,0,0,-8,0,0,0,0,0,0,0,0,0,0,0,
%T A114855 10,0,0,0,0,0,0,-11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,
%U A114855 -14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,-17,0,0,0,0
%N A114855 Expansion of q^(-1/3) * (eta(q) * eta(q^4))^2 / eta(q^2) in powers of q.
%C A114855 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%D A114855 S. Ramanujan, On Certain Arithmetical Functions. Collected Papers of Srinivasa Ramanujan, p. 147, Ed. G. H. Hardy et al., AMS Chelsea 2000.
%D A114855 S. Ramanujan, Notebooks, Tata Institute of Fundamental Research, Bombay 1957 Vol. 1, see page 266.  MR0099904 (20 #6340)
%H A114855 Seiichi Manyama, <a href="/A114855/b114855.txt">Table of n, a(n) for n = 0..10000</a>
%H A114855 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A114855 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H A114855 M. Josuat-Verges and J. S. Kim, <a href="http://arxiv.org/abs/1101.5608">Touchard-Riordan formulas, T-fractions, and Jacobi's triple product identity</a>, p. 28, equation (61), arXiv:1101.5608, 2011
%F A114855 Expansion of psi(x^2) * f(-x)^2 = phi(-x) * f(-x^4)^2 in powers of x where phi(), psi(), f() are Ramanujan theta functions.
%F A114855 Euler transform of period 4 sequence [ -2, -1, -2, -3, ...].
%F A114855 a(n) = b(3*n + 1) where b(n) is multiplicative and a(p^e) = 0 if e is odd, a(3^e) = 0^e, a(p^e) = p^(e/2) if p == 1 (mod 3), a(p^e) = (-p)^(e/2) if p == 2 (mod 3).
%F A114855 Given g.f. A(x), then B(q) = q * A(q^3) satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = (u*w * (u + 2*w) * (u + 4*w))^2 - v^6 * (u^2 + 4*u*w + 8*w^2).
%F A114855 G.f.: Sum_{k} (3*k + 1) * x^(3*k^2 + 2*k) = Product_{k>0} (1 - x^k)^2 * (1 + x^(2*k)) * (1 - x^(4*k)).
%F A114855 a(4*n + 2) = a(4*n + 3) = a(8*n + 4) = 0. a(4*n + 1) = -2 * a(n). 2 * a(n) = A113277(4*n + 1) = - A114855(4*n + 1).
%F A114855 (-1)^n * a(n) = A113277(n). a(8*n) = A080332(n).
%F A114855 G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 6^(3/2) (t/i)^(3/2) f(t) where q = exp(2 Pi i t). - _Michael Somos_, Mar 11 2015
%e A114855 G.f. = 1 - 2*x + 4*x^5 - 5*x^8 + 7*x^16 - 8*x^21 + 10*x^33 - 11*x^40 + ...
%e A114855 G.f. = q - 2*q^4 + 4*q^16 - 5*q^25 + 7*q^49 - 8*q^64 + 10*q^100 - 11*q^121 + ...
%t A114855 a[ n_] := SeriesCoefficient[ QPochhammer[ x^4]^2 EllipticTheta[ 4, 0, x], {x, 0, n}]; (* _Michael Somos_, Mar 11 2015 *)
%t A114855 a[ n_] := SeriesCoefficient[ QPochhammer[ x]^2 EllipticTheta[ 2, 0, x] / (2 x^(1/4)), {x, 0, n}];  (* _Michael Somos_, Mar 11 2015 *)
%t A114855 a[ n_] := With[{m = Sqrt[3 n + 1]}, If[ IntegerQ[ m], -m (-1)^Mod[ m, 3], 0]]; (* _Michael Somos_, Mar 11 2015 *)
%o A114855 (PARI) {a(n) = if( issquare( 3*n + 1, &n), n * -(-1)^(n%3), 0)};
%o A114855 (PARI) {a(n) = my(A, p, e); if( n<0, 0, n = 3*n + 1; A = factor(n); prod( k=1, matsize(A)[1], if( p=A[k,1], e=A[k,2]; if( e%2, 0, (-(-1)^(p%3) * p)^(e/2) )))) };
%o A114855 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A))^2 / eta(x^2 + A), n))};
%o A114855 (Magma) A := Basis( CuspForms( Gamma1(36), 3/2), 300); A[1] - 2*A[4]; /* _Michael Somos_, Mar 11 2015 */
%Y A114855 Cf. A080332, A113277, A114855.
%K A114855 sign
%O A114855 0,2
%A A114855 _Michael Somos_, Jan 01 2006