A185220 Expansion of phi(x^3) * psi(x)^2 / chi(-x) in powers of x where phi(), psi(), chi() are Ramanujan theta functions.

1, 3, 4, 5, 5, 5, 7, 7, 9, 7, 6, 11, 8, 10, 8, 9, 14, 10, 15, 7, 7, 14, 14, 16, 8, 13, 13, 12, 18, 14, 13, 15, 15, 16, 9, 11, 22, 16, 19, 16, 11, 17, 16, 23, 19, 9, 22, 18, 16, 15, 18, 27, 12, 23, 11, 15, 24, 24, 27, 9, 23, 23, 20, 21, 19, 15, 22, 24, 22, 17

Offset: 0

Author

Michael Somos, Aug 29 2013

Keywords

• nonn

Comments

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

Examples

			1 + 3*x + 4*x^2 + 5*x^3 + 5*x^4 + 5*x^5 + 7*x^6 + 7*x^7 + 9*x^8 + 7*x^9 + ...
q^7 + 3*q^31 + 4*q^55 + 5*q^79 + 5*q^103 + 5*q^127 + 7*q^151 + 7*q^175 + ...
		

Links

• Vaclav Kotesovec, 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. A224825, A227595.

Programs

Mathematica

nmax = 100; CoefficientList[Series[Product[(1 - x^k)^2 * (1 + x^k)^5 * (1 - x^(3*k)) / (1 + x^(3*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 08 2015 *)

PARI

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^3 + A)^2 / (eta(x + A)^3 * eta(x^6 + A)), n))}

Formula

Expansion of q^(-7/24) * eta(q^2)^5 * eta(q^3)^2 / (eta(q)^3 * eta(q^6)) in powers of q.

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

G.f.: Product_{k>0} (1 - x^k)^2 * (1 + x^k)^5 * (1 - x^(3*k)) / (1 + x^(3*k)).

a(n) = A224825(3*n) = A227595(3*n).