A035036 Fourier coefficients of E_{gamma,2}*E_{0,4}.

1, 8, -248, 1952, -8440, 25008, -60512, 134464, -270584, 474344, -775248, 1288416, -2059360, 2970352, -4168384, 6101952, -8659192, 11358864, -14704664, 19808800, -26383440, 32809216, -39940896, 51490752, -66022496, 78150008, -92080912, 115265600, -141859520

Offset

0,2

Comments

E_{gamma,2}*E_{0,4} is the unique normalized weight-6 modular form for Gamma_0(2) with an order 1/2 zero at gamma = -1/2 + i/2 and an order 1 zero at 0.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

B. Brent, Quadratic Minima and Modular Forms, Experimental Mathematics, v.7 no.3, 257-274.

H. H. Chan and C. Krattenthaler, Recent progress in the study of representations of integers as sums of squares, arXiv:math/0407061 [math.NT], 2001.

Formula

G.f.: 1 - 8 * Sum_{k>=1} k^5*q^k/(1-(-q)^k).

Expansion of (phi(q)^4 + 16 * q * psi(q^2)^4) * phi(-q)^8 in powers of q where phi(), psi() are Ramanujan theta functions. - Michael Somos, Apr 05 2012

Expansion of (eta(q)^8 + 32 * eta(q^4)^8) * eta(q)^16 / eta(q^2)^12 in powers of q. - Michael Somos, Apr 05 2012

Convolution product of A004011 and A035016. - Michael Somos, Apr 05 2012

Example

G.f. = 1 + 8*q - 248*q^2 + 1952*q^3 - 8440*q^4 + 25008*q^5 - 60512*q^6 + 134464*q^7 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q]^4 + EllipticTheta[ 2, 0, q]^4) EllipticTheta[ 4, 0, q]^8, {q, 0, n}]; (* Michael Somos, Aug 21 2014 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ q]^8 + 32 q QPochhammer[ q^4]^8) QPochhammer[ q]^16 / QPochhammer[ q^2]^12, {q, 0, n}]; (* Michael Somos, Aug 21 2014 *)

Prog

(PARI) {a(n) = if( n<1, n==0, 8 * (sigma(n, 5) - if( n%2, 0, 64 * sigma(n/2, 5))))}; /* Michael Somos, Jul 16 2004 */
(PARI) {a(n) = if( n<1, n==0, polcoeff( 1 - 8 * sum( k=1, n, (-1)^k * k^5 * x^k / (1 - x^k) + x * O(x^n)), n))}; /* Michael Somos, Apr 05 2012 */
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff((eta(x + A)^8 + 32 * x * eta(x^4 + A)^8) * eta(x + A)^16 / eta(x^2 + A)^12, n))}; /* Michael Somos, Apr 05 2012 */
(Magma) A := Basis( ModularForms( Gamma0(2), 6), 29); A[1] + 8*A[2]; /* Michael Somos, Aug 21 2014 */

Crossrefs

Cf. A004011, A035016.

Sequence in context: A299648 A219269 A029751 * A221518 A317519 A300202

Adjacent sequences: A035033 A035034 A035035 * A035037 A035038 A035039

Keyword

easy,sign

Author

Barry Brent (barryb(AT)primenet.com)

Status

approved