A260412 Expansion of psi(x^2) * psi(x^3) / f(-x^2, -x^10) in powers of x where psi(), f(,) are Ramanujan theta functions.

1, 0, 2, 1, 2, 2, 3, 2, 3, 4, 4, 5, 7, 6, 9, 10, 11, 12, 13, 15, 17, 19, 21, 24, 28, 30, 35, 37, 41, 47, 52, 56, 62, 69, 75, 83, 92, 99, 110, 121, 131, 143, 157, 170, 186, 203, 219, 239, 260, 281, 307, 332, 359, 389, 421, 453, 491, 530, 570, 617, 665, 714, 770

Offset: 0

Author

Michael Somos, Jul 24 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^2 + x^3 + 2*x^4 + 2*x^5 + 3*x^6 + 2*x^7 + 3*x^8 + 4*x^9 + ...
G.f. = 1/q + 2*q^47 + q^71 + 2*q^95 + 2*q^119 + 3*q^143 + 2*q^167 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..2500
• 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

Programs

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x] EllipticTheta[ 2, 0, x^(3/2)] / (4 x^(5/8) QPochhammer[ x^2, x^12] QPochhammer[ x^10, x^12] QPochhammer[ x^12]), {x, 0, n}];
nmax = 50; CoefficientList[Series[Product[(1-x^(4*k))^3 * (1-x^(6*k))^3 / ((1-x^(2*k))^2 * (1-x^(3*k)) * (1-x^(12*k))^2), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 11 2016 *)

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A)^3 * eta(x^6 + A)^3 / (eta(x^2 + A)^2 * eta(x^3 + A) * eta(x^12 + A)^2), n))};

Formula

Expansion of q^(1/24) * eta(q^4)^3 * eta(q^6)^3 / (eta(q^2)^2 * eta(q^3) * eta(q^12)^2) in powers of q.

Euler transform of period 12 sequence [ 0, 2, 1, -1, 0, 0, 0, -1, 1, 2, 0, -1, ...].

a(n) ~ exp(Pi*sqrt(n/6)) / (4*sqrt(n)). - Vaclav Kotesovec, Jul 11 2016