A029769 Expansion of eta(q^2)^12 / theta_3(q)^3 in powers of q.

1, -6, 12, -8, 0, 12, -48, 48, -15, 60, -12, -96, 0, -120, 240, 64, 96, -234, -156, 0, 0, 444, -240, -96, -335, 420, 144, 384, 0, -600, -480, -384, 672, -264, 840, 120, 0, -348, 912, -480, -768, -168, -684, 96, 0, 1416, -672, 768, 673, 510, -2328, 0, 0, 144, 1200, 960, -1248, -1332, 1500, -1920

Offset

1,2

Comments

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

References

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).

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).

Links

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Goro Shimura, Modular forms of half-integral weight, Annals of Mathematics, Second Series, Vol. 97, No. 3 (May, 1973), pp. 440-481

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

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

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

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

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

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

a(8*n + 5) = 0. a(4*n) = -8 * a(n). - Michael Somos, Mar 24 2015

Example

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

Maple

with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1, add(add(
      [-9, -6, -3, -6][1+irem(d, 4)]*d, d=divisors(j))*b(n-j), j=1..n)/n)
    end:
a:= n-> b(n-1):
seq(a(n), n=1..50);  # Alois P. Heinz, May 02 2014

Mathematica

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

Prog

(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 */
(Magma) Basis( CuspForms( Gamma0(4), 9/2), 61)[1]; /* Michael Somos, Mar 24 2015 */

Crossrefs

Sequence in context: A103698 A175375 A175365 * A074590 A272966 A105730

Adjacent sequences: A029766 A029767 A029768 * A029770 A029771 A029772

Keyword

sign

Author

N. J. A. Sloane

Status

approved