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%I #23 Mar 12 2021 22:24:44
%S 1,-7,19,-23,6,11,8,-55,73,-42,12,-5,14,-56,114,-119,18,65,20,-138,
%T 152,-84,24,-37,31,-98,235,-184,30,66,32,-247,228,-126,48,49,38,-140,
%U 266,-330,42,88,44,-276,438,-168,48,-101,57,-217,342,-322,54,227,72,-440,380,-210,60,-30,62,-224
%N Expansion of (eta(q) * eta(q^6))^7 / (eta(q^2) * eta(q^3))^5 in powers of q.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%H G. C. Greubel, <a href="/A123532/b123532.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 Euler transform of period 6 sequence [ -7, -2, -2, -2, -7, -4, ...].
%F Expansion of q * (phi(-q) * psi(q^3))^3 / (phi(-q^3) * psi(q)) in powers of q where phi(), psi() are Ramanujan theta functions.
%F Expansion of (b(q)^2 / b(q^2)) * (c(q^2)^2 / c(q)) / 3 in powers of q where b(), c() are cubic AGM theta functions.
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = 6 (t / i)^2 f(t) where q = exp(2 Pi i t). - _Michael Somos_, Sep 19 2013
%F G.f.: x * Product_{k>0} (1 - x^k)^2 * (1 -x^(3*k))^2 * (1 + x^(3*k))^7 / (1 + x^k)^5.
%F Convolution of A006353 and A226235. - _Michael Somos_, Sep 19 2013
%e G.f. = q - 7*q^2 + 19*q^3 -23*q^4 + 6*q^5 + 11*q^6 + 8*q^7 - 55*q^8 + ...
%t a[ n_] := If[ n < 1, 0, Sum[ { 1, -4, 6, -4, 1, 0}[[ Mod[ d, 6, 1]]] d, {d, Divisors[ n]}]]; (* _Michael Somos_, Sep 19 2013 *)
%t a[ n_] := SeriesCoefficient[ q (QPochhammer[ q] QPochhammer[ q^6])^7 / (QPochhammer[ q^2] QPochhammer[ q^3])^5, {q, 0, n}]; (* _Michael Somos_, Sep 19 2013 *)
%o (PARI) {a(n) = if( n<1, 0, sumdiv(n, d, d*[ 0, 1, -4, 6, -4, 1][ d%6 + 1]))};
%o (PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^6 + A))^7 / (eta(x^2 + A) * eta(x^3 + A))^5, n))};
%o (PARI) {a(n) = if( n<1, 0, sumdiv(n, d, n/d *[ 0, 1, -9, 16, -9, 1][ d%6 + 1]))}; /* _Michael Somos_, Sep 19 2013 */
%o (Sage) A = ModularForms( Gamma0(6), 2, prec=50) . basis(); A[1] - 7*A[2]; # _Michael Somos_, Sep 19 2013
%Y Cf. A006353, A226235.
%K sign
%O 1,2
%A _Michael Somos_, Oct 02 2006
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