A266575 Expansion of q * f(-q^4)^6 / phi(-q) in powers of q where phi(), f() are Ramanujan theta functions.

1, 2, 4, 8, 8, 12, 16, 16, 25, 28, 28, 32, 40, 40, 48, 64, 48, 62, 76, 64, 80, 92, 80, 96, 121, 100, 112, 128, 120, 136, 160, 128, 144, 184, 152, 200, 200, 164, 208, 224, 192, 216, 252, 224, 248, 296, 224, 256, 337, 262, 312, 320, 280, 336, 368, 320, 336, 396

Offset: 1

Author

Michael Somos, Jan 03 2016

Keywords

• nonn

Comments

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

Examples

			G.f. = x + 2*x^2 + 4*x^3 + 8*x^4 + 8*x^5 + 12*x^6 + 16*x^7 + 16*x^8 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..2500
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A134414. A245643.

Programs

Magma

A := Basis( ModularForms( Gamma1(4), 5/2), 59); A[2];

Mathematica

a[ n_] := SeriesCoefficient[ q QPochhammer[ q^4]^6 / EllipticTheta[ 4, 0, q], {q, 0, n}];
a[ n_] := SeriesCoefficient[ 2^-4 EllipticTheta[ 3, 0, q] EllipticTheta[ 2, 0, q]^4, {q, 0, n}];

PARI

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^4 + A)^6 / eta(x + A)^2, n))};

Formula

Expansion of q * phi(q) * psi(q^2)^4 in powers of q where phi(), psi() are Ramanujan theta functions.

Expansion of eta(q^2) * eta(q^4)^6 / eta(q)^2 in powers of q.

Euler transform of period 4 sequence [2, 1, 2, -5, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 2^(-3/2) (t/I)^(5/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A245643.

G.f.: x * Product_{k>0} (1 + x^k) * (1 - x^(4*k))^6 / (1 - x^k).

Convolution inverse of A134414.