A228745 Expansion of (phi(q)^4 + 7 * phi(-q)^4) / 8 in powers of q where phi() is a Ramanujan theta function.

1, -6, 24, -24, 24, -36, 96, -48, 24, -78, 144, -72, 96, -84, 192, -144, 24, -108, 312, -120, 144, -192, 288, -144, 96, -186, 336, -240, 192, -180, 576, -192, 24, -288, 432, -288, 312, -228, 480, -336, 144, -252, 768, -264, 288, -468, 576, -288, 96, -342, 744

Offset

0,2

Comments

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

Links

G. C. Greubel, Table of n, a(n) for n = 0..2500

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

a(n) = -6 * b(n) where b() is multiplicative with a(0) = 1, b(2^e) = -4 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)) = 1/2 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A228746.

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

a(2*n) = A004011(n). a(2*n + 1) = -6 * A008438(n).

Convolution with A005875 is A003781.

Example

G.f. = 1 - 6*q + 24*q^2 - 24*q^3 + 24*q^4 - 36*q^5 + 96*q^6 - 48*q^7 + 24*q^8 + ...

Mathematica

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

Prog

(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( (A^4 + 7 * subst(A, x, -x)^4) / 8, n))};
(Magma) A := Basis( ModularForms( Gamma0(4), 2), 51); A[1] - 6*A[2]; /* Michael Somos, Aug 21 2014 */

Crossrefs

Cf. A003781, A004011, A005875, A008438, A228746.

Sequence in context: A280589 A298038 A223751 * A049319 A203984 A237836

Adjacent sequences: A228742 A228743 A228744 * A228746 A228747 A228748

Keyword

sign

Author

Michael Somos, Sep 02 2013

Status

approved