A138520 Expansion of 1 - q * (psi(q^5) / psi(q))^2 in powers of q where psi() is a Ramanujan theta function.

1, -1, 2, -3, 6, -11, 16, -24, 38, -57, 82, -117, 168, -238, 328, -448, 614, -834, 1114, -1480, 1966, -2592, 3384, -4398, 5704, -7361, 9436, -12045, 15344, -19470, 24576, -30922, 38822, -48576, 60548, -75259, 93342, -115454, 142360, -175104, 214958, -263262

Offset

0,3

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 (phi(-q^5) / phi(-q))^2 * (chi^5(-q) / chi(-q^5)) in powers of q where phi(), chi() are Ramanujan theta functions. - Michael Somos, Sep 16 2015

Expansion of (eta(q^5) / eta(q^2))^3 * eta(q) / eta(q^10) in powers of q.

Euler transform of period 10 sequence [ -1, 2, -1, 2, -4, 2, -1, 2, -1, 0, ...].

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u - v)^2 - u * (1 - v) * (5*u - 4).

G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (u - v)^4 - u * (1 - u) * (5*u - 4) * v * (1 - v) * (5*v - 4).

G.f. is a period 1 Fourier series which satisfies f(-1 / (10 t)) = (4/5) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A138522.

G.f.: Product_{k>0} P(5,x^k)^2 / ((1 + x^k)^4 * P(10,x^k)) where P(n,x) is the n-th cyclotomic polynomial.

a(n) = - A138519(n) unless n=0. Convolution inverse of A095813.

a(n) = (-1)^n * A228864(n). - Michael Somos, Sep 16 2015

a(n) ~ (-1)^n * exp(2*Pi*sqrt(n/5)) / (2 * 5^(5/4) * n^(3/4)). - Vaclav Kotesovec, Nov 15 2017

Example

G.f. = 1 - q + 2*q^2 - 3*q^3 + 6*q^4 - 11*q^5 + 16*q^6 - 24*q^7 + 38*q^8 + ...

Mathematica

a[ n_] := SeriesCoefficient[ 1 - (EllipticTheta[ 2, 0, q^(5/2)] / EllipticTheta[ 2, 0, q^(1/2)])^2, {q, 0, n}]; (* Michael Somos, Sep 16 2015 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, q^5] / EllipticTheta[ 4, 0, q])^2 QPochhammer[ -q^5, q^5] / QPochhammer[ -q, q]^5, {q, 0, n}]; (* Michael Somos, Sep 16 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) / eta(x^10 + A) * ( eta(x^5 + A) / eta(x^2 + A) )^3, n))};

Crossrefs

Cf. A095813, A138519, A138522, A228864.

Sequence in context: A102990 A351203 A138519 * A210458 A228864 A289434

Adjacent sequences: A138517 A138518 A138519 * A138521 A138522 A138523

Keyword

sign

Author

Michael Somos, Mar 23 2008

Status

approved