A258040 Expansion of f(x) / f(-x) in powers of x where f() is the g.f. for A007325.

1, -2, 2, 0, -2, 2, 0, 0, -2, 2, 2, -8, 8, 0, -8, 8, -2, 0, -6, 8, 6, -24, 24, 0, -24, 22, -4, 0, -16, 20, 16, -64, 62, 0, -60, 56, -10, 0, -40, 48, 38, -148, 144, 0, -136, 126, -24, 0, -88, 106, 82, -320, 308, 0, -288, 264, -48, 0, -180, 216, 168, -652, 624

Offset

0,2

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

Expansion of f(-x, -x^4) * f(-x^2, +x^3) / (f(+x, -x^4) * f(-x^2, -x^3)) = f(-x, -x^9) * f(+x^3, +x^7) / (f(+x, +x^9) * f(-x^3, -x^7)) in powers of x where f(,) is the Ramanujan general theta function.

Euler transform of period 20 sequence [ -2, 1, 2, 0, 0, -1, 2, 0, -2, 0, -2, 0, 2, -1, 0, 0, 2, 1, -2, 0, ...].

a(10*n + 3) = a(10*n + 7) = 0.

Example

G.f. = 1 - 2*x + 2*x^2 - 2*x^4 + 2*x^5 - 2*x^8 + 2*x^9 + 2*x^10 - 8*x^11 + ...

Mathematica

a[ n_] := SeriesCoefficient[ Product[(1 - x^k)^{ 2, -1, -2, 0, 0, 1, -2, 0, 2, 0, 2, 0, -2, 1, 0, 0, -2, -1, 2, 0}[[ Mod[k, 20, 1]]], {k, 1, n}], {x, 0, n}];

Prog

(PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^ [ 0, 2, -1, -2, 0, 0, 1, -2, 0, 2, 0, 2, 0, -2, 1, 0, 0, -2, -1, 2][k%20 + 1]) , n))};

Crossrefs

Cf. A007325.

Sequence in context: A214665 A352557 A229723 * A215879 A114700 A353768

Adjacent sequences: A258037 A258038 A258039 * A258041 A258042 A258043

Keyword

sign

Author

Michael Somos, May 16 2015

Status

approved