A247223 Expansion of f(-x^5, -x^7) in powers of x where f() is a Ramanujan theta function.

1, 0, 0, 0, 0, -1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Author

Michael Somos, Jan 10 2015

Keywords

• sign

Examples

			G.f. = 1 - x^5 - x^7 + x^22 + x^26 - x^51 - x^57 + x^92 + x^100 - x^145 + ...
G.f. = q - q^121 - q^169 + q^529 + q^625 - q^1225 - q^1369 + q^2209 + ...
		

Links

• Seiichi Manyama, 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

Programs

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[q^5,q^12]*QPochhammer[q^7,q^12] *QPochhammer[q^12,q^12], {q, 0, n}]; (* G. C. Greubel, Dec 08 2017 *)

PARI

{a(n) = my(m = 24*n + 1); if( issquare(m, &m) && (m%12==1 || m%12==11), (-1)^((m+6) \ 12))};

Formula

Euler transform of period 12 sequence [ 0, 0, 0, 0, -1, 0, -1, 0, 0, 0, 0, -1, ...]. - Michael Somos, Jan 10 2015

G.f.: Product_{k>0} (1 - x^(12*k)) * (1 - x^(12*k - 5)) * (1 - x^(12*k - 7)).