A261240 Expansion of f(-x^6, -x^12)^2 / (f(-x, -x) * f(-x^3, -x^15)) in powers of x where f(, ) is Ramanujan's general theta function.

1, 2, 4, 9, 16, 28, 47, 76, 120, 185, 280, 416, 608, 878, 1252, 1765, 2464, 3408, 4676, 6364, 8600, 11545, 15400, 20424, 26938, 35346, 46152, 59981, 77616, 100016, 128369, 164140, 209120, 265510, 335992, 423840, 533035, 668404, 835804, 1042308, 1296448

Offset: 0

Author

Michael Somos, Aug 12 2015

Keywords

• nonn

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			G.f. = 1 + 2*x + 4*x^2 + 9*x^3 + 16*x^4 + 28*x^5 + 47*x^6 + 76*x^7 + ...
G.f. = q^-1 + 2*q + 4*q^3 + 9*q^5 + 16*q^7 + 28*q^9 + 47*q^11 + ...
		

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], Sep 30 2015.
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A058647, A186115, A186964, A187020.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ x^(3/4) QPochhammer[ x^6] EllipticTheta[ 2, 0, x^(3/2)] / (EllipticTheta[ 4, 0, x] EllipticTheta[ 2, 0, x^(9/2)]), {x, 0, n}];
nmax=60; CoefficientList[Series[Product[(1+x^k) * (1-x^(6*k))^3 * (1-x^(9*k)) / ((1-x^k) * (1-x^(3*k)) * (1-x^(18*k))^2),{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Oct 13 2015 *)

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^6 + A)^3 * eta(x^9 + A) / (eta(x + A)^2 * eta(x^3 + A) * eta(x^18 + A)^2), n))};

Formula

Expansion of f(-x^6) * psi(x^3) / (phi(-x) * psi(x^9)) in powers of x where phi(), psi(), f() are Ramanujan theta functions.

Expansion of q^(1/2) * eta(q^2) * eta(q^6)^3 * eta(q^9) / (eta(q)^2 * eta(q^3) * eta(q^18)^2) in powers of q.

Euler transform of period 18 sequence [2, 1, 3, 1, 2, -1, 2, 1, 2, 1, 2, -1, 2, 1, 3, 1, 2, 0, ...].

a(n) = A058647(2*n - 1) = A186115(2*n - 1) = A186964(2*n - 1) = A187020(2*n - 1).

a(n) ~ exp(2*Pi*sqrt(2*n)/3) / (2^(7/4) * sqrt(3) * n^(3/4)). - Vaclav Kotesovec, Oct 13 2015