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URL: https://oeis.org/A035036

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A035036
Fourier coefficients of E_{gamma,2}*E_{0,4}.
3
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
Barry 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
Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = (3/8) * Pi^3 / Gamma(3/4)^12 = A388402. - Simon Plouffe, Sep 15 2025
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
Sequence in context: A299648 A219269 A029751 * A221518 A317519 A300202
KEYWORD
easy,sign
AUTHOR
Barry Brent (barryb(AT)primenet.com)
STATUS
approved