A260599 Expansion of psi(x^4) / chi(-x)^2 in powers of x where psi(), chi() are Ramanujan theta functions.

1, 2, 3, 6, 10, 16, 25, 38, 55, 80, 115, 160, 223, 306, 415, 560, 747, 988, 1301, 1700, 2206, 2850, 3661, 4676, 5950, 7536, 9500, 11936, 14936, 18620, 23141, 28662, 35386, 43566, 53480, 65466, 79937, 97356, 118277, 143370, 173391, 209232, 251966, 302806

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

Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], 2015-2016.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of f(-x^4)^4 / (f(-x)^2 * phi(x^2)) in powers of x where phi(), f() are Ramanujan theta functions.

Expansion of q^(-7/12) * eta(q^2)^2 * eta(q^8)^2 / (eta(q)^2 * eta(q^4)) in powers of q.

Euler transform of period 8 sequence [ 2, 0, 2, 1, 2, 0, 2, -1, ...].

2 * a(n) = A260574(4*n + 2).

a(n) ~ exp(sqrt(2*n/3)*Pi) / (8*sqrt(2*n)). - Vaclav Kotesovec, Oct 14 2015

Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = (1/8) * exp(7*Pi/12) * Gamma(5/8) * sqrt(2) / Pi^(1/4) / Gamma(7/8) = A388923. - Simon Plouffe, Sep 21 2025

Example

G.f. = 1 + 2*x + 3*x^2 + 6*x^3 + 10*x^4 + 16*x^5 + 25*x^6 + 38*x^7 + ...
G.f. = q^7 + 2*q^19 + 3*q^31 + 6*q^43 + 10*q^55 + 16*q^67 + 25*q^79 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x]^2 EllipticTheta[ 2, 0, x^2] / (2 x^(1/2)), {x, 0, n}];
a[ n_] := SeriesCoefficient[ QPochhammer[ x^4]^4 / (QPochhammer[ x]^2 EllipticTheta[ 3, 0, x^2]), {x, 0, n}];
nmax=60; CoefficientList[Series[Product[(1+x^k)^2 * (1-x^(8*k))^2 / (1-x^(4*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 14 2015 *)

Prog

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

Crossrefs

Cf. A260574.

Sequence in context: A075623 A024801 A324742 * A280908 A146163 A101277

Adjacent sequences: A260596 A260597 A260598 * A260600 A260601 A260602

Keyword

nonn

Author

Michael Somos, Jul 29 2015

Status

approved