A260676 Expansion of phi(x) * psi(x^30) in powers of x where phi(), chi() are Ramanujan theta functions.

1, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 1, 2, 0, 0, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0

Offset: 0

Author

Michael Somos, Nov 14 2015

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 + 2*x^4 + 2*x^9 + 2*x^16 + 2*x^25 + x^30 + 2*x^31 + 2*x^34 + ...
G.f. = q^15 + 2*q^19 + 2*q^31 + 2*q^51 + 2*q^79 + 2*q^115 + q^135 + 2*q^139 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A260671.

Programs

Magma

m:=130;
f:= func< q | (&*[( (1-q^(2*n))^5*(1-q^(60*n))^2 )/( (1-q^n)^2*(1-q^(4*n))^2*(1-q^(30*n)) ): n in [1..m+1]]) >;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( f(x) )); // G. C. Greubel, Feb 02 2023

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[3, 0, x] QPochhammer[x^60]^2 / QPochhammer[x^30], {x, 0, n}];

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^60 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^30 + A)), n))};

SageMath

m = 130
def f(q): return product( ((1-q^(2*n))^5*(1-q^(60*n))^2)/((1-q^n)^2*(1-q^(4*n))^2*(1-q^(30*n))) for n in range(1,m+2))
def A260676_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( f(x) ).list()
A260676_list(m) # G. C. Greubel, Feb 02 2023

Formula

Expansion of q^(-15/4) * eta(q^2)^5 * eta(q^60)^2 / (eta(q)^2 * eta(q^4)^2 * eta(q^30)) in powers of q.

Euler transform of a period 60 sequence.

2 * a(n) = A260671(4*n + 15).