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

0,2

Comments

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

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

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).

Example

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 + ...

Mathematica

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

Prog

(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))};
(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<x>:=PowerSeriesRing(Integers(), m);
Coefficients(R!( f(x) )); // G. C. Greubel, Feb 02 2023
(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.<x> = PowerSeriesRing(ZZ, prec)
    return P( f(x) ).list()
A260676_list(m) # G. C. Greubel, Feb 02 2023

Crossrefs

Cf. A260671.

Sequence in context: A318985 A033749 A033743 * A033741 A033735 A033731

Adjacent sequences: A260673 A260674 A260675 * A260677 A260678 A260679

Keyword

nonn

Author

Michael Somos, Nov 14 2015

Status

approved