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A113277 Expansion of q^(-1/3) * eta(q^2)^5 / eta(q)^2 in powers of q.

%I A113277 #30 Jun 08 2025 15:19:52
%S A113277 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 A113277 -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 A113277 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 A113277 Expansion of q^(-1/3) * eta(q^2)^5 / eta(q)^2 in powers of q.
%C A113277 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A113277 Seiichi Manyama, <a href="/A113277/b113277.txt">Table of n, a(n) for n = 0..10000</a>
%H A113277 Y. Martin and K. Ono, <a href="http://dx.doi.org/10.1090/S0002-9939-97-03928-2">Eta-quotients and elliptic curves</a>, Proc. Amer. Math. Soc. 125 (1997), no. 11, 3169-3176. MR1401749 (97m:11057)
%H A113277 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A113277 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A113277 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(2^e) = -(-2)^(e/2), a(p^e) = p^(e/2) if p == 1 (mod 3), a(p^e) = (-p)^(e/2) if p == 2 (mod 3).
%F A113277 Euler transform of period 2 sequence [ 2, -3, ...].
%F A113277 G.f.: Sum_{k} (3*k + 1) * (-x)^(3*k^2 + 2*k) = Product_{k>0} (1 - x^k)^3 * (1 + x^k)^5.
%F A113277 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 A113277 G.f. is a period 1 Fourier series which satisfies f(-1 / (48 t)) = 3456^(1/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is g.f. for A080332.
%F A113277 a(4*n + 2) = a(4*n + 3) = a(5*n + 2) = a(5*n + 4) = a(8*n + 4) = 0. a(25*n + 8) = -5 * a(n). A114855(n) = (-1)^n * a(n). a(4*n + 1) = 2 * A114855(n). a(8*n) = A080332(n).
%F A113277 G.f.: Product_{n >= 1} (1 - q^(4*n))^3 * (1 + q^(4*n-1))^2 * (1 - q^(4*n-2)) * (1 + q^(4*n-3))^2 = Product_{n >= 1} (1 - q^(4*n))^3 * (1 - q^(4*n-1))^(-2) * (1 - q^(4*n-2))^3 * (1 - q^(4*n-3))^(-2). - _Peter Bala_, Jun 07 2025
%e A113277 G.f. = 1 + 2*x - 4*x^5 - 5*x^8 + 7*x^16 + 8*x^21 - 10*x^33 - 11*x^40 + 13*x^56 + ...
%e A113277 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 A113277 QP = QPochhammer; s = QP[q^2]^5/QP[q]^2 + O[q]^100; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 25 2015 *)
%o A113277 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 / eta(x + A)^2, n))};
%o A113277 (PARI) {a(n) = if(issquare( 3*n + 1, &n), n * (-1)^(n%3 + n), 0)};
%o A113277 (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==2) * (-(-1)^(p%3) * p)^(e/2)))))};
%Y A113277 Cf. A080332, A114855.
%K A113277 sign
%O A113277 0,2
%A A113277 _Michael Somos_, Oct 21 2005