A035016 -id:A035016 - cp's OEIS frontend

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

Barry Brent (barryb(AT)primenet.com)

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.

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

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.

Cf. A004011, A035016.

A := Basis( ModularForms( Gamma0(2), 6), 29); A[1] + 8*A[2]; /* Michael Somos, Aug 21 2014 */

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 *)

{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 */

{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 */

{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 */

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

A259491 Expansion of (eta(q)^2 * eta(q^2) * eta(q^4)^3 / eta(q^8)^2)^2 in powers of q.

Original entry on oeis.org

1, -4, 0, 16, -16, 8, 0, -96, 112, 44, 0, 176, -448, -88, 0, -32, 1136, -200, 0, -176, -2016, 384, 0, 224, 3136, 484, 0, -608, -5504, -792, 0, 640, 9328, -704, 0, 192, -12112, 648, 0, 352, 14112, 792, 0, -208, -21312, -88, 0, -2112, 31808, -932, 0, 800

Offset: 0

Michael Somos, Jun 28 2015

sign

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

			G.f. = 1 - 4*q + 16*q^3 - 16*q^4 + 8*q^5 - 96*q^7 + 112*q^8 + 44*q^9 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A030211, A035016, A131999.

A := Basis( ModularForms( Gamma1(8), 4), 52); A[1] - 4*A[2] + 16*A[4] - 16*A[5] + 8*A[6] - 96*A[8] + 112*A[9];

a[ n_] := SeriesCoefficient[ (QPochhammer[ q]^2 QPochhammer[ q^2] QPochhammer[ q^4]^3 / QPochhammer[ q^8]^2)^2, {q, 0, n}];
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^2] EllipticTheta[ 4, 0, q]^2)^2, {q, 0, n}];
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, q] EllipticTheta[ 4, 0, q^2] EllipticTheta[ 4, 0, q^4]^2)^2, {q, 0, n}];

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^2 * eta(x^2 + A) * eta(x^4 + A)^3 / eta(x^8 + A)^2)^2, n))};

Expansion of (phi(q) * phi(q^2) * phi(-q)^2)^2 in powers of q where phi() is a Ramanujan theta function.
Euler transform of period 8 sequence [ -4, -6, -4, -12, -4, -6, -4, -8, ...].
G.f.: Product_{k>0} ((1 - x^k)^4 * (1 + x^k)^2 * (1 + x^(2*k)) / (1 + x^(4*k))^2)^2.
a(2*n + 1) = -4 * A030211(n). a(4*n) = A035016(n). a(4*n + 2) = 0.
Convolution square of A131999.

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

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A259491 Expansion of (eta(q)^2 * eta(q^2) * eta(q^4)^3 / eta(q^8)^2)^2 in powers of q.

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Magma

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