%I #22 Mar 12 2021 22:24:43
%S 1,-6,21,-68,198,-510,1248,-2904,6393,-13604,28044,-55956,108982,
%T -207552,386622,-707216,1271970,-2250582,3925780,-6757272,11483232,
%U -19290824,32057352,-52722744,85884503,-138644292,221885805,-352241792,554892894
%N Expansion of q / (chi(q) * chi(q^3))^6 in powers of q where chi() is a Ramanujan theta function.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A107653/b107653.txt">Table of n, a(n) for n = 1..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 Convolution inverse of A186829. - _Michael Somos_, Feb 27 2011
%F Expansion of (eta(q) * eta(q^3) * eta(q^4) * eta(q^12) / (eta(q^2) * eta(q^6))^2)^6 in powers of q.
%F Euler transform of period 12 sequence [-6, 6, -12, 0, -6, 12, -6, 0, -12, 6, -6, 0, ...]. - _Michael Somos_, Jun 13 2005
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = f(t) where q = exp(2 Pi i t). - _Michael Somos_, Feb 27 2011
%F G.f.: x * (Product_{k>0} ((1 + x^(2*k)) * (1 + x^(6*k))) / ((1 + x^k) * (1 + x^(3*k))))^6 = x * (Product_{k>0} (1 + x^(2*k-1)) * (1 + x^(6*k-3)))^-6.
%F a(n) = -(-1)^n * A123653(n). - _Michael Somos_, Feb 27 2011
%e G.f. = q - 6*q^2 + 21*q^3 - 68*q^4 + 198*q^5 - 510*q^6 + 1248*q^7 + ...
%t QP = QPochhammer; s = (QP[q]*QP[q^3]*QP[q^4]*(QP[q^12]/(QP[q^2]*QP[q^6])^2 ))^6 + O[q]^30; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 14 2015, adapted from PARI *)
%t a[n_]:= SeriesCoefficient[q/( QPochhammer[-q, q^2]* QPochhammer[-q^3, q^6])^6, {q, 0, n}]; Table[a[n], {n, 1, 50}] (* _G. C. Greubel_, Dec 09 2017 *)
%o (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^3 + A) * eta(x^4 + A) * eta(x^12 + A) / (eta(x^2 + A)^2 * eta(x^6 + A)^2))^6, n))}; /* _Michael Somos_, Jun 13 2005 */
%Y Cf. A123653, A186829.
%K sign
%O 1,2
%A _N. J. A. Sloane_, Jun 07 2005
%E Revised by _Michael Somos_, Jun 12 2005