%I #23 Jan 29 2024 01:22:38
%S 1,1,-1,1,-1,-4,-1,-6,-1,1,4,12,-1,14,6,-4,-1,-16,-1,-18,4,-6,-12,24,
%T -1,21,-14,1,6,-28,4,-30,-1,12,16,24,-1,38,18,14,4,-40,6,-42,-12,-4,
%U -24,48,-1,43,-21,-16,-14,-52,-1,-48,6,-18,28,60,4,62,30,-6,-1
%N Expansion of f(x) * f(x^3) * f(-x^4)^2 * chi(-x^6)^2 in powers of x where chi(), f() are Ramanujan theta functions.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A257656/b257656.txt">Table of n, a(n) for n = 0..10000</a>
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>, 2019.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F Expansion of eta(q^2)^3 * eta(q^4) * eta(q^6)^5 / (eta(q) * eta(q^3) * eta(q^12)^3) in powers of q.
%F Euler transform of period 12 sequence [1, -2, 2, -3, 1, -6, 1, -3, 2, -2, 1, -4, ...].
%F a(n) = a(3*n) = (-1)^n * A109039(n). a(2*n) = A109039(n).
%F Multiplicative with a(2^e) = -1, a(p^e) = ((p*Kronecker(12, p))^(e+1) - 1)/(p*Kronecker(12, p) - 1) for odd prime p. - _Andrew Howroyd_, Jul 27 2018
%F Sum_{k=1..n} abs(a(k)) ~ c * n^2, where c = Pi^2/(24*sqrt(3)) = 0.237425... . - _Amiram Eldar_, Jan 29 2024
%e G.f. = 1 + x - x^2 + x^3 - x^4 - 4*x^5 - x^6 - 6*x^7 - x^8 + x^9 + 4*x^10 + ...
%t a[ n_] := SeriesCoefficient[ QPochhammer[ -x] QPochhammer[ -x^3] QPochhammer[ x^4]^2 QPochhammer[ x^6, x^12]^2, {x, 0, n}];
%t a[ n_] := If[ n<1, Boole[n == 0], Times @@ (If[ # < 5, -(-1)^#, With[ {t = # KroneckerSymbol[ 12, #]}, (t^(#2 + 1) - 1 ) / (t - 1)]]& @@@ FactorInteger @ n)];
%o (PARI) {a(n) = if( n<1, n==0, -sumdiv(n, d, d * kronecker( 12, d) * (-1)^(n/d)))};
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^4 + A) * eta(x^6 + A)^5 / (eta(x + A) * eta(x^3 + A) * eta(x^12 + A)^3), n))};
%o (PARI) {a(n) = my(A, p, e, t); if( n<1, n==0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p<5, -(-1)^p, t = p * kronecker( 12, p); (t^(e+1) - 1) / (t - 1))))};
%Y Cf. A109039.
%Y Cf. A000122, A000700, A010054, A121373.
%K sign,mult
%O 0,6
%A _Michael Somos_, Jul 25 2015