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%I #46 Sep 08 2022 08:44:50
%S 1,-2,-2,4,0,4,0,-8,-5,0,14,-8,0,0,0,16,2,10,-34,0,0,-28,0,16,25,0,28,
%T 0,0,0,0,-32,-28,-4,0,-20,0,68,0,0,-46,0,14,56,0,0,0,-32,49,-50,-4,0,
%U 0,-56,0,0,68,0,-82,0,0,0,0,64,0,56,62,8,0,0,0,40,-142,0,-50,-136,0,0,0,0,-11,92,158,0,0,-28,0
%N Expansion of eta(q)^2 * eta(q^2) * eta(q^4) * eta(q^8)^2 in powers of q.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C Unique cusp form of weight 3 for congruence group Gamma_1(8). - _Michael Somos_, Aug 11 2011
%C Associated with permutations in Mathieu group M24 of shape (8)^2(4)(2)(1)^2.
%C For n nonzero, a(n) is nonzero if and only if n is in A002479.
%C Number 20 of the 74 eta-quotients listed in Table I of Martin (1996).
%H Seiichi Manyama, <a href="/A030207/b030207.txt">Table of n, a(n) for n = 1..10000</a>
%H M. Koike, <a href="http://projecteuclid.org/euclid.nmj/1118787564">On McKay's conjecture</a>, Nagoya Math. J., 95 (1984), 85-89.
%H M. Koike, <a href="http://projecteuclid.org/euclid.nmj/1118787871">Mathieu group M24 and modular forms</a>, Nagoya Math. J., 99 (1985), 147-157. MR0805086 (87e:11060)
%H Y. Martin, <a href="http://dx.doi.org/10.1090/S0002-9947-96-01743-6">Multiplicative eta-quotients</a>, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
%H Michael Somos, <a href="/A030203/a030203.txt">Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers</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 q * phi(q) * phi(-q)^2 * phi(q^2) * psi(q^4)^2 in powers of q where phi(), psi() are Ramanujan theta functions. - _Michael Somos_, May 28 2007
%F Expansion of (3 * phi(q)^3 * phi(q^2)^3 - 2 * phi(q) * phi(q^2)^5 - phi(q)^5 * phi(q^2)) / 4 in powers of q where phi() is a Ramanujan theta function. - _Michael Somos_, Jun 13 2007
%F Euler transform of period 8 sequence [-2, -3, -2, -4, -2, -3, -2, -6, ...]. - _Michael Somos_, May 28 2007
%F a(n) is multiplicative with a(2^e) = (-2)^e, a(p^e) = (1+(-1)^e)/2 * p^e if p == 5, 7 (mod 8), a(p^e) = a(p)*a(p^(e-1)) - p^2*a(p^(e-2)) if p == 1, 3 (mod 8) where a(p) = 4*x^2 -2*p and p = x^2 + 2*y^2. - _Michael Somos_, Jun 13 2007
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 512^(1/2) (t/i)^3 f(t) where q = exp(2 Pi i t). - _Michael Somos_, Jul 25 2007
%F G.f.: (1/2) * Sum_{u,v in Z} (u*u - 2*v*v) * x^(u*u + 2*v*v). - _Michael Somos_, Jun 14 2007
%F G.f.: x * Product_{k>0} (1 - x^k)^6 * (1 + x^k)^4 * (1 + x^(2*k))^3 * (1 + x^(4*k))^6. - _Michael Somos_, May 28 2007
%F a(8*n + 5) = a(8*n + 7) = 0. a(2*n) = -2*a(n). a(8*n + 1) = A128712(n). a(8*n + 3) = -2 * A128713(n).
%e G.f. = q - 2*q^2 - 2*q^3 + 4*q^4 + 4*q^6 - 8*q^8 - 5*q^9 + 14*q^11 - 8*q^12 + ...
%t a[ n_] := SeriesCoefficient[ q QPochhammer[ q]^2 QPochhammer[ q^2] QPochhammer[ q^4] QPochhammer[ q^8]^2, {q, 0, n}]; (* _Michael Somos_, Aug 11 2011 *)
%t a[ n_] := SeriesCoefficient[ (1/4) EllipticTheta[ 3, 0, q] EllipticTheta[ 4, 0, q]^2 EllipticTheta[ 3, 0, q^2] EllipticTheta[ 2, 0, q^2]^2, {q, 0, n}]; (* _Michael Somos_, May 17 2015 *)
%t a[ n_] := SeriesCoefficient[ (-EllipticTheta[ 3, 0, q]^5 EllipticTheta[ 3, 0, q^2] + 3 EllipticTheta[ 3, 0, q]^3 EllipticTheta[ 3, 0, q^2]^3 - 2 EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^2]^5) / 4, {q, 0, n}]; (* _Michael Somos_, May 17 2015 *)
%o (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^8 + A))^2 * eta(x^2 + A) * eta(x^4 + A), n))}; /* _Michael Somos_, May 28 2007 */
%o (PARI) {a(n) = my(A, p, e, x, y, a0, a1); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, (-2)^e, p%8>4, if( e%2, 0, p^e), for( x=1, sqrtint(p\2), if( issquare( p - 2*x^2, &y), break)); y = 4*y^2 - 2*p; a0=1; a1=y; for( i=2, e, x = y*a1 - p^2*a0; a0=a1; a1=x); a1)))}; /* _Michael Somos_, Jun 13 2007 */
%o (Sage) CuspForms( Gamma1(8), 3, prec = 100).0; # _Michael Somos_, Aug 11 2011
%o (Magma) Basis( CuspForms( Gamma1(8), 3), 100) [1]; /* _Michael Somos_, May 27 2014 */
%Y Cf. A002479, A128712, A128713.
%K sign,mult
%O 1,2
%A _N. J. A. Sloane_.
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