A134747 Expansion of q * (chi(-q) / chi(-q^4))^8 in powers of q where chi() is a Ramanujan theta function.

1, -8, 28, -64, 142, -352, 792, -1536, 2917, -5744, 10868, -19200, 33414, -58816, 101256, -167936, 275314, -452392, 732748, -1160064, 1819808, -2851104, 4421064, -6752256, 10236407, -15476272, 23215192, -34450944, 50811638, -74701632, 109138272, -158171136

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..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of k * (1 - k) / ( 4 * (1 + k) ) in powers of q^(1/2) where q is Jacobi's nome and k is the elliptic modulus.

Expansion of ( (eta(q) * eta(q^8)) / (eta(q^2) * eta(q^4)) )^8 in powers of q.

Euler transform of period 8 sequence [ -8, 0, -8, 8, -8, 0, -8, 0, ...].

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = 16 * u*w * (v*w-1) * (v*u-1) - (v - u^2) * (v - w^2).

G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = f(t) where q = exp(2 Pi i t).

G.f.: x * ( Product_{k>0} (1 + x^(4*k)) / (1 + x^k) )^8.

Convolution inverse of A131123.

Convolution 8th power of A261734. - Michael Somos, Oct 16 2015

a(n) ~ -(-1)^n * exp(Pi*sqrt(2*n)) / (2^(21/4) * n^(3/4)). - Vaclav Kotesovec, Apr 10 2018

Empirical: Sum_{n>=1} a(n)/exp(2*Pi*n) = -8 - 6*sqrt(2) + (3/2)*sqrt(60 + 43*sqrt(2)). - Simon Plouffe, Mar 04 2021

Example

G.f. = q - 8*q^2 + 28*q^3 - 64*q^4 + 142*q^5 - 352*q^6 + 792*q^7 - 1536*q^8 + ...

Mathematica

a[ n_] := SeriesCoefficient[ q (QPochhammer[ q, q^2] QPochhammer[ -q^4, q^4])^8, {q, 0, n}]; (* Michael Somos, Oct 16 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( ( (eta(x + A) * eta(x^8 + A)) / (eta(x^2 + A) * eta(x^4 + A)) )^8, n))};

Crossrefs

Cf. A131123, A261734.

Sequence in context: A353325 A101127 A007259 * A083013 A350144 A028553

Adjacent sequences: A134744 A134745 A134746 * A134748 A134749 A134750

Keyword

sign

Author

Michael Somos, Nov 07 2007

Status

approved