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

%I A035036 #25 Sep 24 2022 01:44:07
%S A035036 1,8,-248,1952,-8440,25008,-60512,134464,-270584,474344,-775248,
%T A035036 1288416,-2059360,2970352,-4168384,6101952,-8659192,11358864,
%U A035036 -14704664,19808800,-26383440,32809216,-39940896,51490752,-66022496,78150008,-92080912,115265600,-141859520
%N A035036 Fourier coefficients of E_{gamma,2}*E_{0,4}.
%C A035036 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.
%H A035036 G. C. Greubel, <a href="/A035036/b035036.txt">Table of n, a(n) for n = 0..1000</a>
%H A035036 B. Brent, <a href="http://www.emis.de/journals/EM/expmath/volumes/7/7.html">Quadratic Minima and Modular Forms</a>, Experimental Mathematics, v.7 no.3, 257-274.
%H A035036 H. H. Chan and C. Krattenthaler, <a href="https://arxiv.org/abs/math/0407061">Recent progress in the study of representations of integers as sums of squares</a>, arXiv:math/0407061 [math.NT], 2001.
%F A035036 G.f.: 1 - 8 * Sum_{k>=1} k^5*q^k/(1-(-q)^k).
%F A035036 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
%F A035036 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
%F A035036 Convolution product of A004011 and A035016. - _Michael Somos_, Apr 05 2012
%e A035036 G.f. = 1 + 8*q - 248*q^2 + 1952*q^3 - 8440*q^4 + 25008*q^5 - 60512*q^6 + 134464*q^7 + ...
%t A035036 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 *)
%t A035036 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 *)
%o A035036 (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 */
%o A035036 (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 */
%o A035036 (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 */
%o A035036 (Magma) A := Basis( ModularForms( Gamma0(2), 6), 29); A[1] + 8*A[2]; /* _Michael Somos_, Aug 21 2014 */
%Y A035036 Cf. A004011, A035016.
%K A035036 easy,sign
%O A035036 0,2
%A A035036 Barry Brent (barryb(AT)primenet.com)