A228746 Expansion of 8 * phi(q)^4 - 7 * phi(-q)^4 in powers of q where phi() is a Ramanujan theta function.

1, 120, 24, 480, 24, 720, 96, 960, 24, 1560, 144, 1440, 96, 1680, 192, 2880, 24, 2160, 312, 2400, 144, 3840, 288, 2880, 96, 3720, 336, 4800, 192, 3600, 576, 3840, 24, 5760, 432, 5760, 312, 4560, 480, 6720, 144, 5040, 768, 5280, 288, 9360, 576, 5760, 96, 6840

Offset: 0

Author

Michael Somos, Sep 02 2013

Keywords

• nonn
• easy

Comments

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

Convolution with A005875 is A004008.

Examples

			G.f. = 1 + 120*q + 24*q^2 + 480*q^3 + 24*q^4 + 720*q^5 + 96*q^6 + 960*q^7 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..2500
• Michael Somos, Introduction to Ramanujan theta functions, 2019.
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Crossrefs

Cf. A004008, A004011, A005875, A008438, A212002, A228745.

Cf. A000122, A000700, A010054, A121373.

Programs

Magma

A := Basis( ModularForms( Gamma0(4), 2), 50); A[1] + 120*A[2]; /* Michael Somos, Aug 21 2014 */

Mathematica

a[ n_] := SeriesCoefficient[ (8 EllipticTheta[ 3, 0, q]^4 - 7 EllipticTheta[ 4, 0, q]^4), {q, 0, n}];

PARI

{a(n) = my(A); if( n<0, 0, A = sum(k=1, sqrtint(n), 2 * x^k^2, 1 + x * O(x^n)); polcoeff( 8 * A^4 - 7 * subst(A, x, -x)^4, n))};

Formula

a(n) = 120 * b(n) with b() multiplicative where b(2^e) = 1/5 if e>1, b(p^e) = b(p) * b(p^(e-1)) - p * b(p^(e-2)), if p>2.

G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 32 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A228745.

G.f.: 8 * (Sum_{k in Z} x^k^2)^4 - 7 * (Sum_{k in Z} (-x)^k^2)^4 .

a(2*n) = A004011(n). a(2*n + 1) = 120 * A008438(n).

Sum_{k=1..n} a(k) ~ c * n^2, where c = 4*Pi^2 = 39.478417... (A212002). - Amiram Eldar, Dec 29 2023