A239051 Expansion of (f(-q^2, -q^3)^5 - 3 * q * f(-q, -q^4)^5) / f(-q)^3 in powers of q where f() is a Ramanujan theta function.

1, 0, 10, -10, 10, 0, 0, 10, 0, -10, 10, 0, 10, -10, 20, -10, 0, 10, -10, 0, 10, 0, 20, -10, 0, 0, 0, 0, 10, 0, 0, 0, 10, -20, 20, 10, 0, 10, 0, -20, 0, 0, 20, -10, 20, -10, 0, 10, -10, 10, 10, 0, 10, -10, 0, 0, 0, 0, 0, 0, 10, 0, 20, -10, 10, -10, 0, 10, 10

Offset

0,3

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..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Moebius transform is period 5 sequence [ 0, 10, -10, 0, 0, ...].

G.f.: 1 + 10 * ( Sum_{k>=0} x^(5*k + 2) / (1 - x^(5*k + 2)) - x^(5*k + 3) / (1 - x^(5*k + 3)) ).

a(n) = A227216(n) - 3 * A229802(n).

a(5*n) = a(n). a(5*n + 1) = 0.

Example

G.f. = 1 + 10*q^2 - 10*q^3 + 10*q^4 + 10*q^7 - 10*q^9 + 10*q^10 + 10*q^12 + ...

Mathematica

a[ n_] := If[ n < 1, Boole[ n == 0], 10 Sum[ {0, 1, -1, 0, 0}[[ Mod[ d, 5, 1] ]] ], {d, Divisors @ n}]];

Prog

(PARI) {a(n) = if( n<1, n==0, 10 * sumdiv(n, d, (d%5==2) - (d%5==3)))};
(Sage) ModularForms( Gamma1(5), 1, prec=70).0;
(Magma) Basis( ModularForms( Gamma1(5), 1), 70) [1];

Crossrefs

Cf. A227216, A229802.

Sequence in context: A087028 A145279 A103708 * A131722 A072803 A163139

Adjacent sequences: A239048 A239049 A239050 * A239052 A239053 A239054

Keyword

sign

Author

Michael Somos, Jun 13 2014

Status

approved