A213419 Expansion of q * chi(-q) / chi(-q^25) in powers of q where chi() is a Ramanujan theta function.

1, -1, 0, -1, 1, -1, 1, -1, 2, -2, 2, -2, 3, -3, 3, -4, 5, -5, 5, -6, 7, -8, 8, -9, 11, -11, 11, -14, 15, -16, 17, -19, 22, -23, 24, -27, 31, -32, 34, -38, 42, -44, 47, -52, 57, -61, 64, -70, 78, -82, 87, -96, 103, -110, 117, -127, 138, -146, 155, -168, 182

Offset

1,9

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 (eta(q) * eta(q^50)) / (eta(q^2) * eta(q^25)) in powers of q.

Euler transform of period 50 sequence [ -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, 0, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (50 t)) = f(t) where q = exp(2 Pi i t).

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

G.f.: x * (Product_{k>0} (1 + x^(25*k)) / (1 + x^k)).

Convolution inverse of A034320.

a(n) ~ -(-1)^n * exp(2*Pi*sqrt(n)/5) / (2*sqrt(5)*n^(3/4)). - Vaclav Kotesovec, Jun 06 2018

Example

G.f. = q - q^2 - q^4 + q^5 - q^6 + q^7 - q^8 + 2*q^9 - 2*q^10 + 2*q^11 - 2*q^12 + ...

Mathematica

QP = QPochhammer; s = (QP[q]*QP[q^50])/(QP[q^2]*QP[q^25]) + O[q]^70; CoefficientList[s, q] (* Jean-François Alcover, Nov 14 2015, adapted from PARI *)
a[ n_] := SeriesCoefficient[ q QPochhammer[ -q^25, q^25] / QPochhammer[ -q, q], {q, 0, n}]; (* Michael Somos, May 05 2016 *)

Prog

(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^50 + A) / (eta(x^2 + A) * eta(x^25 + A)), n))};

Crossrefs

Cf. A034320.

Sequence in context: A270433 A169994 A169995 * A000700 A081362 A112216

Adjacent sequences: A213416 A213417 A213418 * A213420 A213421 A213422

Keyword

sign

Author

Michael Somos, Jun 11 2012

Status

approved