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

1,2

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 = 1..2500

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

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.

Example

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

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}];

Prog

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

Crossrefs

Cf. A134414. A245643.

Sequence in context: A360156 A054785 A236924 * A260514 A123263 A008218

Adjacent sequences: A266572 A266573 A266574 * A266576 A266577 A266578

Keyword

nonn

Author

Michael Somos, Jan 03 2016

Status

approved