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A173763 Expansion of (eta(q^2)^7 / eta(q^4)^2)^4 + 16 * (eta(q)^2 * eta(q^2) * eta(q^4)^2)^4 in powers of q.

%I A173763 #31 Aug 06 2025 01:06:59
%S A173763 1,16,-156,256,870,-2496,-952,4096,4653,13920,-56148,-39936,178094,
%T A173763 -15232,-135720,65536,-247662,74448,315380,222720,148512,-898368,
%U A173763 204504,-638976,-1196225,2849504,2344680,-243712,-3840450,-2171520,-1309408,1048576,8759088,-3962592,-828240,1191168,4307078
%N A173763 Expansion of (eta(q^2)^7 / eta(q^4)^2)^4 + 16 * (eta(q)^2 * eta(q^2) * eta(q^4)^2)^4 in powers of q.
%C A173763 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A173763 G. C. Greubel, <a href="/A173763/b173763.txt">Table of n, a(n) for n = 1..1000</a>
%H A173763 LMFDB, <a href="http://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/2/10/a/a/">Newform orbit 2.10.a.a</a>.
%H A173763 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A173763 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A173763 Expansion of q * (psi(q)^3 * phi(-q)^2)^4 * ((phi(q) / psi(q))^4 + 16 * q * (psi(q) / phi(q))^4) in powers of q where phi(), psi() are Ramanujan theta functions.
%F A173763 a(n) is multiplicative with a(2^e) = 16^e, a(p^e) = a(p) * a(p^(e-1)) - p^9 * a(p^(e-2)) if p>2.
%F A173763 G.f. is a period 1 Fourier series which satisfies f(-1 / (2 t)) = 32 (t / i)^10 f(t) where q = exp(2 Pi i t).
%e A173763 G.f. = q + 16*q^2 - 156*q^3 + 256*q^4 + 870*q^5 - 2496*q^6 - 952*q^7 + 4096*q^8 + ...
%t A173763 a[ n_] := SeriesCoefficient[ q (QPochhammer[ q^2]^7 / QPochhammer[ q^4]^2)^4 + 16 q^2 (QPochhammer[ q]^2 QPochhammer[ q^2] QPochhammer[ q^4]^2)^4, {q, 0, n}]; (* _Michael Somos_, May 28 2013 *)
%o A173763 (PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^2 + A)^7 / eta(x^4 + A)^2)^4 + 16 * x * (eta(x + A)^2 * eta(x^2 + A) * eta(x^4 + A)^2)^4, n))};
%o A173763 (PARI) q='q+O('q^99); Vec((eta(q^2)^7/eta(q^4)^2)^4+16*q*(eta(q)^2*eta(q^2)*eta(q^4)^2)^4) \\ _Altug Alkan_, Apr 18 2018
%o A173763 (Sage) CuspForms( Gamma1(2), 10, prec=50).0; # _Michael Somos_, May 28 2013
%o A173763 (Magma) Basis( CuspForms( Gamma1(2), 10), 50) [1]; /* _Michael Somos_, May 27 2014 */
%Y A173763 Cf. A002288.
%K A173763 sign,mult
%O A173763 1,2
%A A173763 _Michael Somos_, Feb 23 2010