A229802 Expansion of q * f(-q, -q^4)^5 / f(-q)^3 in powers of q where f() is a Ramanujan theta function.

1, -2, 4, -3, 1, 2, -2, 0, 3, -2, 2, -2, 4, -6, 4, 1, -2, 4, 0, -3, 2, -4, 4, 0, 1, 2, 0, -4, 0, 2, 2, -2, 8, -6, -2, 1, -2, 0, 6, 0, 2, -4, 4, -6, 3, 2, -2, 4, -3, -2, 2, -2, 4, 0, 2, 0, 0, 0, 0, -2, 2, -4, 4, -3, 4, 4, -2, -4, 6, -6, 2, 0, 4, -6, 4, 0, -4

Offset

1,2

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 = 1..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q * f(-q)^2 * (f(-q^5) / f(-q^2, -q^3))^5 = q * f(-q, -q^4)^2 * (f(-q^5) / f(-q^2, -q^3))^3 in powers of q where f() is a Ramanujan theta function. - Michael Somos, Jun 10 2014

Euler transform of period 5 sequence [ -2, 3, 3, -2, -2, ...].

Moebius transform is period 5 sequence [ 1, -3, 3, -1, 0, ...]. - Michael Somos, Jun 10 2014

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

a(5*n) = a(n).

Example

G.f. = q - 2*q^2 + 4*q^3 - 3*q^4 + q^5 + 2*q^6 - 2*q^7 + 3*q^9 - 2*q^10 + ...

Mathematica

a[ n_] := SeriesCoefficient[ q QPochhammer[ q]^2 / (QPochhammer[ q^2, q^5] QPochhammer[ q^3, q^5])^5, {q, 0, n}]; (* Michael Somos, Jun 10 2014 *)
a[ n_] := If[ n < 1, 0, Sum[ Im[(I - 3) {1, I, -I, -1, 0}[[ Mod[ d, 5, 1] ]] ], {d, Divisors @ n}]]; (* Michael Somos, Jun 10 2014 *)

Prog

(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( prod(k=1, n, (1 - x^k + A)^[ 2, 2, -3, -3, 2][k%5 + 1]), n))};
(PARI) {a(n) = if( n<1, 0, sumdiv(n, d, imag( (I - 3) * [ 0, 1, I, -I, -1][ d%5 + 1])))}; /* Michael Somos, Jun 10 2014 */
(Sage) ModularForms( Gamma1(5), 1, prec=70).1;
(Magma) Basis( ModularForms( Gamma1(5), 1), 70) [2] ; /* Michael Somos, Jun 10 2014 */

Crossrefs

Cf. A227216.

Sequence in context: A285587 A134893 A227862 * A106581 A317612 A070402

Adjacent sequences: A229799 A229800 A229801 * A229803 A229804 A229805

Keyword

sign

Author

Michael Somos, Sep 29 2013

Status

approved