A201077 G.f.: 1 / Product_{i>=1} (1-q^(2*i-1))^2*(1-q^(12*i-8))*(1-q^(12*i-6))*(1-q^(12*i-4))*(1-q^(12*i)).

1, 2, 3, 6, 10, 16, 26, 40, 60, 90, 131, 188, 269, 378, 525, 726, 993, 1346, 1816, 2430, 3230, 4274, 5619, 7348, 9570, 12400, 15994, 20554, 26303, 33530, 42602, 53934, 68053, 85614, 107370, 134262, 167443, 208250, 258329, 319680

Offset: 0

Author

N. J. A. Sloane, Nov 26 2011

Keywords

• nonn

Comments

The expansion of the denominator is 1, -2, 1, -2, 3, -2, 3, -2, 3, -4, 4, -6, 4, -6, 7, -8, 9, -8, 11, -10, 12, -12, 13, -16, 16, -20, 21,... - R. J. Mathar, Nov 27 2011

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

Examples

			G.f. = 1 + 2*x + 3*x^2 + 6*x^3 + 10*x^4 + 16*x^5 + 26*x^6 + 40*x^7 + 60*x^8 + ...
G.f. = q + 2*q^7 + 3*q^13 + 6*q^19 + 10*q^25 + 16*q^31 + 26*q^37 + 40*q^43 + ...
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Márton Balázs, Dan Fretwell, and Jessica Jay, Interacting Particle Systems and Jacobi style identities, arXiv:2011.05006 [math.PR], 2020.
• Brian Drake, Limits of areas under lattice paths, Discrete Math. 309 (2009), no. 12, 3936-3953.
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Programs

Mathematica

max = 39; den[i_] := Series[(1-q^(2*i-1))^2*(1-q^(12*i-8))*(1-q^(12*i-6))*(1-q^(12*i-4))*(1-q^(12*i)), {q, 0, max }] // Normal; gf = 1/Product[den[i], {i, 1, max}]; Series[gf, {q, 0, max}] // CoefficientList[#, q]& (* Jean-François Alcover, Mar 18 2014 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] / (QPochhammer[ x] QPochhammer[ x^6, x^12]), {x, 0, n}]; (* Michael Somos, Feb 18 2017 *)

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^12 + A) / (eta(x + A)^2 * eta(x^4 + A) * eta(x^6 + A)), n))} /* Michael Somos, Jun 07 2012 */

Formula

Expansion of chi(x) / (f(-x) * chi(-x^6)) in powers of x where chi(), f() are Ramanujan theta functions. - Michael Somos, Jun 07 2012

Expansion of q^(-1/6) * eta(q^2)^2 * eta(q^12) / (eta(q)^2 * eta(q^4) * eta(q^6)) in powers of q. - Michael Somos, Jun 07 2012

Expansion of f(x^1, x^5) / (f(-x, -x^2) * f(-x^3, -x^6)) in powers of x where f(, ) is Ramanujan's general theta function. - Michael Somos, Feb 18 2017

Euler transform of period 12 sequence [2, 0, 2, 1, 2, 1, 2, 1, 2, 0, 2, 1, ...]. - Michael Somos, Feb 18 2017

a(n) ~ exp(2^(3/2)*Pi*sqrt(n)/3) / (3*2^(3/2)*n). - Vaclav Kotesovec, Jun 28 2025