A138504 Expansion of (eta(q^2)^9 / (eta(q)^2 * eta(q^4)^4))^2 in powers of q.

1, 4, -4, -32, -4, 104, 32, -192, -4, 292, -104, -480, 32, 680, 192, -832, -4, 1160, -292, -1440, -104, 1536, 480, -2112, 32, 2604, -680, -2624, 192, 3368, 832, -3840, -4, 3840, -1160, -4992, -292, 5480, 1440, -5440, -104, 6728, -1536, -7392, 480, 7592, 2112, -8832, 32, 9412, -2604, -9280

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..10000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

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

Euler transform of period 4 sequence [ 4, -14, 4, -6, ...].

a(n) = 4 * b(n) where a(0) = 1, b(n) is multiplicative with b(2^e) = -1 if e>0, b(p^e) = ((p^2)^(e+1) - 1) / (p^2 - 1) if p == 1 (mod 4), b(p^e) = ((-p^2)^(e+1) - 1) / ( -p^2 - 1) if p == 3 (mod 4).

G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 32 (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A122854.

G.f.: 1 + 4 * Sum_{k>0} -(-1)^k * (2*k-1)^2 * x^(2*k-1) / (1 + x^(2*k-1)).

a(n) = (-1)^n * A120030(n). a(n) = 4 * A138505(n) unless n=0.

Example

G.f. = 1 + 4*q - 4*q^2 - 32*q^3 - 4*q^4 + 104*q^5 + 32*q^6 - 192*q^7 - 4*q^8 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (QPochhammer[ q^2]^9 / (QPochhammer[ q]^2 QPochhammer[ q^4]^4))^2, {q, 0, n}]; (* Michael Somos, May 24 2015 *)
a[ n_] := If[ n < 1, Boole[n == 0], -4 DivisorSum[ n, #^2 KroneckerSymbol[ -4, #] (-1)^(n/#) &]]; (* Michael Somos, May 24 2015 *)

Prog

(PARI) {a(n) = if( n<1, n==0, -4 * sumdiv(n, d, d^2 * kronecker(-4, d) * (-1)^(n/d)))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^9 / (eta(x + A)^2 * eta(x^4 + A)^4))^2, n))};

Crossrefs

Cf. A120030, A138505.

Sequence in context: A196131 A256691 A120030 * A002611 A130188 A270676

Adjacent sequences: A138501 A138502 A138503 * A138505 A138506 A138507

Keyword

sign

Author

Michael Somos, Mar 21 2008

Status

approved