A262090 Expansion of f(x^3, x^21) / f(-x^2, -x^4) where f(, ) is the Ramanujan general theta function.

1, 0, 1, 1, 2, 1, 3, 2, 5, 3, 7, 5, 11, 7, 15, 11, 22, 15, 30, 22, 42, 31, 56, 43, 77, 58, 101, 80, 135, 106, 177, 142, 232, 187, 299, 246, 388, 319, 495, 415, 634, 532, 803, 683, 1017, 869, 1277, 1103, 1605, 1390, 2000, 1751, 2492, 2189, 3087, 2733, 3819

Offset: 0

Author

Michael Somos, Sep 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^2 + x^3 + 2*x^4 + x^5 + 3*x^6 + 2*x^7 + 5*x^8 + 3*x^9 + ...
G.f. = q^77 + q^173 + q^221 + 2*q^269 + q^317 + 3*q^365 + 2*q^413 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..2500
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A143067.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ -x^3, x^24] QPochhammer[ -x^21, x^24] QPochhammer[ x^24] / QPochhammer[ x^2], {x, 0, n}];

PARI

{a(n) = if( n<0, 0, A = x * O(x^n); polcoeff( subst( prod(k=1, n\3, 1 - x^k * [1, 1, 0, 0, 0, 0, 0, 1][k%8 + 1], 1 + x * O(x^(n\3))), x, -x^3) / eta(x^2 + x * O(x^n)), n))};

Formula

Euler transform of period 48 sequence [ 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, ...].

a(n) = - A143067(2*n + 3).