%I #21 Sep 08 2022 08:46:12
%S 1,-2,3,-4,5,-8,7,-8,9,-10,14,-12,16,-14,15,-20,17,-18,19,-24,26,-22,
%T 23,-28,25,-32,32,-28,29,-30,38,-32,33,-40,40,-44,42,-38,39,-40,57,
%U -42,43,-44,45,-62,47,-56,49,-56,62,-52,53,-60,64,-68,64,-58,59,-60
%N Expansion of (psi(-x^3) * f(-x, x^2))^2 in powers of x where psi(), 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="/A258831/b258831.txt">Table of n, a(n) for n = 0..1000</a>
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F Expansion of (f(-x^6)^2 * chi(x^3) / chi(x))^2 in powers of x where chi(), f() are Ramanujan theta functions.
%F Expansion of q^(-5/6) * (eta(q) * eta(q^4) * eta(q^6)^4 / (eta(q^2)^2 * eta(q^3) * eta(q^12)))^2 in powers of q.
%F Euler transform of period 12 sequence [-2, 2, 0, 0, -2, -4, -2, 0, 0, 2, -2, -4, ...].
%F a(n) = (-1)^n * A098098(n) = A208435(2*n + 1) = A208457(2*n + 1). 6 * a(n) = A121613(3*n + 2).
%F Convolution square of A258832.
%e G.f. = 1 - 2*x + 3*x^2 - 4*x^3 + 5*x^4 - 8*x^5 + 7*x^6 - 8*x^7 + 9*x^8 + ...
%e G.f. = q^5 - 2*q^11 + 3*q^17 - 4*q^23 + 5*q^29 - 8*q^35 + 7*q^41 - 8*q^47 + ...
%p with(numtheory):
%p seq((-1)^(n-1)*sigma(6*n - 1)/6, n=1..10^3); # _Muniru A Asiru_, Jan 30 2018
%t a[ n_] := If[ n < 0, 0, (-1)^n DivisorSigma[ 1, 6 n + 5] / 6];
%t a[ n_] := SeriesCoefficient[ (QPochhammer[ x^6]^2 QPochhammer[ x, -x] / QPochhammer[ x^3, -x^3])^2, {x, 0, n}];
%t Table[(-1)^n DivisorSigma[1, 6 n + 5] / 6, {n, 0, 60}] (* _Vincenzo Librandi_, Jan 30 2018 *)
%o (PARI) {a(n) = if(n<0, 0, (-1)^n*sigma(6*n+5)/6)};
%o (PARI) {a(n) = my(A); if(n<0, 0, A = x*O(x^n); polcoeff((eta(x + A)*eta(x^4 + A)*eta(x^6 + A)^4/(eta(x^2+A)^2*eta(x^3+A)*eta(x^12+A)))^2, n))};
%o (GAP) List([0..70], n -> (-1)^n*Sigma(6*n+5)/6); # _Muniru A Asiru_, Jan 30 2018
%o (Magma) [(-1)^n*SumOfDivisors(6*n+5)/6: n in [0..70]]; // _Vincenzo Librandi_, Jan 30 2018
%Y Cf. A098098, A121613, A208435, A208457, A258832.
%K sign
%O 0,2
%A _Michael Somos_, Jun 11 2015