A030204 - OEIS

%I #45 Sep 08 2022 08:44:50

%S 1,-1,-2,1,0,2,1,0,0,-2,1,-2,-2,0,2,-1,0,2,0,2,0,1,0,0,-2,0,0,0,-1,-2,

%T -2,0,2,0,0,-2,3,0,0,2,0,0,2,0,2,-1,-2,0,0,0,-2,2,0,-2,-2,-1,-2,2,0,0,

%U 0,0,0,0,0,2,1,0,0,0,0,2,2,0,2,-2,0,-2,1,0

%N Expansion of q^(-1/8) * eta(q) * eta(q^2) in powers of q.

%C Number 66 of the 74 eta-quotients listed in Table I of Martin (1996).

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%C A030204, A083650 and A138514 are the same except for signs. - _N. J. A. Sloane_, May 07 2010

%H Seiichi Manyama, <a href="/A030204/b030204.txt">Table of n, a(n) for n = 0..10000</a>

%H S. R. Finch, <a href="http://arXiv.org/abs/math.NT/0701251">Powers of Euler's q-Series</a>, arXiv:math/0701251 [math.NT], 2007.

%H M. Koike, <a href="http://projecteuclid.org/euclid.nmj/1118787564">On McKay's conjecture</a>, Nagoya Math. J., 95 (1984), 85-89.

%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 J. B. Tunnell, <a href="http://dx.doi.org/10.1007/BF01389327">A classical Diophantine problem and modular forms of weight 3/2</a>, Invent. Math., 72 (1983), 323-334. [a(n) = coefficient of q^(9m+1) in the q-expansion of the unique normalized newform g of weight 1, level 128, and character chi_2. - _N. J. A. Sloane_, Oct 18 2014]

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F G.f.: Product_{k>0} (1 - x^k) * (1 - x^(2*k)).

%F G.f.: (Sum_{k>0} x^((k^2 - k)/2)) * (Sum_{k in Z} (-1)^k * x^k^2). - _Michael Somos_, Sep 02 2006

%F Expansion of psi(x) * phi(-x) = f(-x^2) * f(-x) = f(-x)^2 / chi(-x) = f(-x)^3 / phi(-x) = f(-x^2)^2 * chi(-x) = f(-x^2)^3 / psi(x) = psi(-x) * phi(-x^2) = psi(x)^2 * chi(-x)^3 = phi(-x)^2 / chi(-x)^3 = (f(-x)^3 * psi(x))^(1/2) = (f(-x^2)^3 * phi(-x))^(1/2) in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions. - _Michael Somos_, Mar 22 2008

%F Expansion of psi(x) * psi(-x) in powers of x^2 where psi() is a Ramanujan theta function. - _Michael Somos_, Oct 11 2013

%F Euler transform of period 2 sequence [ -1, -2, ...].

%F a(3*n) = A107063(n). a(3*n + 2) = -2 * A107064(n). - _Michael Somos_, Oct 11 2013

%F a(9*n + 1) = -a(n), a(9*n + 4) = a(9*n + 7) = 0. - _Michael Somos_, Mar 17 2004

%F a(n) = b(8*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = 0 if p === 3,5,7 (mod 8) and e odd, b(p^e) = (-1)^(e/2) if p == 3 (mod 8) and e even, b(p^e) = 1 if p == 5,7 (mod 8) and e even, b(p^e) = e + 1 if p == 1 (mod 8) and p = x^2 + 32*y^2, b(p^e) = (-1)^e * (e + 1) if p == 1 (mod 8) and p is not of the form x^2 + 32*y^2.

%F a(n) = (-1)^n * A138514(n). Convolution inverse is A002513.

%F G.f.: exp(Sum_{k>=1} (sigma(2*k) - 4*sigma(k))*x^k/k). - _Ilya Gutkovskiy_, Sep 19 2018

%e G.f. = 1 - x - 2*x^2 + x^3 + 2*x^5 + x^6 - 2*x^9 + x^10 - 2*x^11 - 2*x^12 + ...

%e G.f. = q - q^9 - 2*q^17 + q^25 + 2*q^41 + q^49 - 2*q^73 + q^81 - 2*q^89 - 2*q^97 + ...

%t a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ x^2], {x, 0, n}]; (* _Michael Somos_, Oct 11 2013 *)

%t a[ n_] := SeriesCoefficient[ QPochhammer[ x]^2 / QPochhammer[ x, x^2], {x, 0, n}]; (* _Michael Somos_, Oct 11 2013 *)

%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x] EllipticTheta[ 2, 0, x^(1/2)] / (2 x^(1/8)), {x, 0, n}]; (* _Michael Somos_, Oct 11 2013 *)

%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x] EllipticTheta[ 2, 0, I x] / (4 Sqrt[ x] I^(1/4)), {x, 0, 4 n}]; (* _Michael Somos_, Oct 11 2013 *)

%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x] EllipticTheta[ 2, Pi/4, x] / (2^(3/2) x^(1/2)), {x, 0, 4 n}]; (* _Michael Somos_, Jan 31 2015 *)

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^2 + A), n))};

%o (PARI) {a(n) = my(A, p, e); if( n<0, 0, n = 8*n + 1; A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p==2, 0, p%8==1, (e + 1) * if( qfbclassno(-4*p)%8, (-1)^e, 1), e%2==0, (-1)^(e/2*(p%8<5)))))}; /* _Michael Somos_, Jul 26 2006 */

%o (PARI) {a(n) = if( n<0, 0, n = 8*n + 1; (qfrep([1, 0;0, 32], n) - qfrep([4, 2; 2, 9], n))[n])}; /* _Michael Somos_, Sep 02 2006 */

%o (Magma) Basis( CuspForms( Gamma1(128), 1), 641)[1]; /* _Michael Somos_, Jan 31 2015 */

%Y Cf. A000203, A002513, A107063, A107064, A138514.

%K sign

%O 0,3

%A _N. J. A. Sloane_.