%I #30 Sep 08 2022 08:44:50

%S 1,-6,12,-8,0,12,-48,48,-15,60,-12,-96,0,-120,240,64,96,-234,-156,0,0,

%T 444,-240,-96,-335,420,144,384,0,-600,-480,-384,672,-264,840,120,0,

%U -348,912,-480,-768,-168,-684,96,0,1416,-672,768,673,510,-2328,0,0,144,1200,960,-1248,-1332,1500,-1920

%N Expansion of eta(q^2)^12 / theta_3(q)^3 in powers of q.

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

%D Lomadze, G.A.: On the representations of natural numbers by sums of nine squares. Acta. Arith. 68(3), 245-253 (1994). (Russian). See Equation (3.10).

%D Goro Shimura, Modular forms of half-integral weight, pp. 57-74 of Modular Functions of One Variable I (Antwerp 1972), Lect. Notes Math. 320 (1973).

%H Alois P. Heinz, <a href="/A029769/b029769.txt">Table of n, a(n) for n = 1..10000</a>

%H Goro Shimura, <a href="http://www.jstor.org/stable/1970831">Modular forms of half-integral weight</a>, Annals of Mathematics, Second Series, Vol. 97, No. 3 (May, 1973), pp. 440-481

%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 * f(-q^2)^12 / phi(q)^3 in powers of q where phi(), f() are Ramanujan theta functions. - _Michael Somos_, Mar 24 2015

%F Expansion of eta(q)^6 * eta(q^4)^6 / eta(q^2)^3 in powers of q.

%F Euler transform of period 4 sequence [ -6, -3, -6, -9, ...].

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 2^(9/2) (t/i)^(9/2) f(t) where q = exp(2 Pi i t). - _Michael Somos_, Jul 25 2007

%F G.f.: x * (Product_{k>0} (1 - x^k)^2 * (1 + x^(2*k)) * (1 - x^(4*k)))^3.

%F a(8*n + 5) = 0. a(4*n) = -8 * a(n). - _Michael Somos_, Mar 24 2015

%e G.f. = q - 6*q^2 + 12*q^3 - 8*q^4 + 12*q^6 - 48*q^7 + 48*q^8 - 15*q^9 + ...

%p with(numtheory):

%p b:= proc(n) option remember; `if`(n=0, 1, add(add(

%p [-9, -6, -3, -6][1+irem(d, 4)]*d, d=divisors(j))*b(n-j), j=1..n)/n)

%p end:

%p a:= n-> b(n-1):

%p seq(a(n), n=1..50); # _Alois P. Heinz_, May 02 2014

%t b[n_] := b[n] = If[n == 0, 1, Sum[Sum[{-9, -6, -3, -6}[[1+Mod[d, 4]]]*d, {d, Divisors[j]}]*b[n-j], {j, 1, n}]/n]; a[n_] := b[n-1]; Table[a[n], {n, 1, 60}] (* _Jean-François Alcover_, Oct 07 2014, after _Alois P. Heinz_ *)

%t a[ n_] := SeriesCoefficient[ q QPochhammer[ q^2]^12 / EllipticTheta[ 3, 0, q]^3, {q, 0, n}]; (* _Michael Somos_, Mar 24 2015 *)

%o (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A)^6 * eta(x^4 + A)^6 / eta(x^2 + A)^3, n))}; /* _Michael Somos_, Apr 24 2004 */

%o (Magma) Basis( CuspForms( Gamma0(4), 9/2), 61)[1]; /* _Michael Somos_, Mar 24 2015 */

%K sign

%O 1,2

%A _N. J. A. Sloane_