A260183 Expansion of f(x, x^2) * f(x^4, x^8) / f(-x^3, -x^6)^2 in powers of x where f(, ) is Ramanujan's general theta function.

1, 1, 1, 2, 3, 4, 6, 8, 10, 14, 18, 23, 30, 38, 47, 60, 74, 91, 114, 139, 169, 207, 250, 301, 364, 436, 520, 622, 739, 875, 1038, 1224, 1439, 1694, 1985, 2321, 2714, 3162, 3677, 4275, 4956, 5735, 6634, 7655, 8819, 10155, 11669, 13389, 15354, 17575, 20091

Offset: 0

Author

Michael Somos, Nov 10 2015

Keywords

• nonn

Comments

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

Examples

			G.f. = 1 + x + x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 6*x^6 + 8*x^7 + 10*x^8 + 14*x^9 + ...
G.f. = 1/q + q + q^3 + 2*q^5 + 3*q^7 + 4*q^9 + 6*q^11 + 8*q^13 + 10*q^15 + ...
		

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

Crossrefs

Cf. A003105.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x^2, x^12] QPochhammer[ x^10, x^12] QPochhammer[ x^12, x^24] QPochhammer[ x^8] / QPochhammer[x], {x, 0, n}];

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^8 + A) * eta(x^12 + A)^2 / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A) * eta(x^24 + A)), n))};

Formula

Expansion of q^(1/2) * eta(q^2) * eta(q^8) * eta(q^12)^2 / (eta(q) * eta(q^4) * eta(q^6) * eta(q^24)) in powers of q.

Euler transform of period 24 sequence [ 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, ...].

a(n) = A003105(2*n + 1).