A291124 Expansion of phi(x)^6 * phi(-x)^2 in powers of x where phi() is a Ramanujan theta function.

1, 8, 16, -32, -144, -16, 448, 192, -912, -88, 2016, -352, -4032, 176, 5504, 64, -7056, 400, 12112, 352, -18144, -768, 21312, -448, -25536, -968, 35168, 1216, -49536, 1584, 56448, -1280, -56208, 1408, 78624, -384, -109008, -1296, 109760, -704, -114912, -1584

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..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of (eta(q^2)^7 / (eta(q)^2 * eta(q^4)^3))^4 in powers of q.

Euler transform of period 4 sequence [8, -20, 8, -8, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 512 (t/i)^4 g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A045820.

G.f.: Product_{k>0} (1 - x^(2*k))^28 / ((1 - x^k)^8 * (1 - x^(4*k))^12).

a(2*n + 1) = 8 * A030211(n). a(4*n + 2) = 16 * A045823(n).

a(2*n) = 16 * (-1)^n * (-sigma_3(n) + sigma_3(n/4)) where sigma_3(n) is the sum of the cubes of the divisors of n if n is an integer else 0.

Convolution square of A207541.

Example

G.f. = 1 + 8*x + 16*x^2 - 32*x^3 - 144*x^4 - 16*x^5 + 448*x^6 + 192*x^7 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x]^6 EllipticTheta[ 4, 0, x]^2, {x, 0, n}];
a[ n_] := SeriesCoefficient[ (QPochhammer[x^2]^7 / (QPochhammer[ x]^2 QPochhammer[ x^4]^3))^4, {x, 0, n}];

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^7 / (eta(x + A)^2 * eta(x^4 + A)^3))^4, n))};
(PARI) lista(nn) = {q='q+O('q^nn); Vec((eta(q^2)^7/(eta(q)^2*eta(q^4)^3))^4)} \\ Altug Alkan, Mar 21 2018
(Magma) A := Basis( ModularForms( Gamma0(16), 4), 42); A[1] + 8*A[2] + 16*A[3] - 32*A[4] - 144*A[5] - 16*A[6] + 448*A[7] + 192*A[8] - 912*A[9];

Crossrefs

Cf. A030211, A045820, A045823, A207541.

Sequence in context: A146541 A363014 A261976 * A239280 A043111 A043891

Adjacent sequences: A291121 A291122 A291123 * A291125 A291126 A291127

Keyword

sign

Author

Michael Somos, Aug 17 2017

Status

approved