A213617 Expansion of psi(x) * f(-x^3)^3 in powers of x where psi() and f() are Ramanujan theta functions.

1, 2, 3, 3, 3, 5, 4, 5, 4, 5, 7, 5, 8, 4, 5, 8, 8, 9, 5, 7, 9, 6, 9, 9, 7, 10, 10, 11, 5, 6, 12, 12, 10, 10, 7, 10, 12, 14, 10, 5, 15, 8, 13, 8, 12, 17, 10, 16, 7, 9, 14, 12, 15, 11, 11, 12, 12, 16, 14, 15, 13, 15, 13, 7, 12, 17, 16, 15, 10, 13, 18, 16, 20

Offset: 0

Author

Michael Somos, Jun 16 2012

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 + 3*x^2 + 3*x^3 + 3*x^4 + 5*x^5 + 4*x^6 + 5*x^7 + 4*x^8 + 5*x^9 + ...
G.f. = q^11 + 2*q^35 + 3*q^59 + 3*q^83 + 3*q^107 + 5*q^131 + 4*q^155 + 5*q^179 + ...
		

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. A213618.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x^2]^2 QPochhammer[ x^3]^3 / QPochhammer[x]^2, {x, 0, n}]; (* Michael Somos, Apr 26 2015 *)

PARI

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

Formula

Expansion of q^(-11/24) * eta(q^2)^2 * eta(q^3)^3 / eta(q)^2 in powers of q.

Euler transform of period 6 sequence [ 2, 0, -1, 0, 2, -3, ...].

6 * a(n) = A213618(24*n + 11).