A260518 Expansion of psi(x)^2 * f(-x^3)^3 / f(-x) in powers of x where psi(), f() are Ramanujan theta functions.

1, 3, 5, 7, 8, 11, 13, 14, 17, 16, 21, 23, 25, 27, 21, 32, 33, 35, 37, 32, 42, 38, 45, 47, 40, 51, 56, 55, 50, 48, 61, 63, 64, 70, 56, 62, 73, 80, 77, 63, 81, 83, 74, 87, 72, 91, 98, 95, 104, 64, 101, 103, 105, 107, 88, 112, 98, 115, 114, 112, 121, 123, 125

Offset: 0

Author

Michael Somos, Jul 28 2015

Keywords

• nonn

Comments

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

Examples

			G.f. = 1 + 3*x + 5*x^2 + 7*x^3 + 8*x^4 + 11*x^5 + 13*x^6 + 14*x^7 + ...
G.f. = q^7 + 3*q^19 + 5*q^31 + 7*q^43 + 8*q^55 + 11*q^67 + 13*q^79 + ...
		

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

Programs

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x^2]^4 QPochhammer[ x^3]^3 / QPochhammer[ x]^3, {x, 0, n}];

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^4 * eta(x^3 + A)^3 / eta(x + A)^3, n))};

PARI

my(q='q+O('q^99)); Vec(eta(q^2)^4*eta(q^3)^3/eta(q)^3) \\ Altug Alkan, Aug 01 2018

Formula

Expansion of psi(x) * psi(x^3) * f(x, x^2)^2 in powers of x where psi(), f() are Ramanujan theta functions.

Expansion of q^(-7/12) * eta(q^2)^4 * eta(q^3)^3 / eta(q)^3 in powers of q.

Euler transform of period 6 sequence [ 3, -1, 0, -1, 3, -4, ...].

G.f.: Product_{k>0} (1 - x^(2*k))^4 * (1 + x^k + x^(2*k))^3.