A132212 Expansion of f(-x, -x^7) / f(-x, -x) in powers of q where f(, ) is Ramanujan's general theta function.

1, 1, 2, 4, 6, 10, 16, 23, 34, 50, 71, 100, 140, 192, 262, 356, 476, 634, 840, 1102, 1440, 1872, 2417, 3108, 3980, 5070, 6434, 8135, 10242, 12852, 16076, 20036, 24898, 30852, 38112, 46958, 57708, 70730, 86486, 105508, 128412, 155952, 189004, 228580

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

Euler transform of period 8 sequence [ 1, 1, 2, 1, 2, 1, 1, 0, ...].

Example

G.f. = 1 + x + 2*x^2 + 4*x^3 + 6*x^4 + 10*x^5 + 16*x^6 + 23*x^7 + 34*x^8 + 50*x^9 + ...
G.f. = q^9 + q^25 + 2*q^41 + 4*q^57 + 6*q^73 + 10*q^89 + 16*q^105 + 23*q^121 + ...

Mathematica

a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^-{ 1, 1, 2, 1, 2, 1, 1, 0}[[Mod[k, 8, 1]]], {k, n}], {x, 0, n}]; (* Michael Somos, Nov 01 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^8] QPochhammer[ x^7, x^8] QPochhammer[ x^8] / EllipticTheta[ 4, 0, x], {x, 0, n}]; (* Michael Somos, Nov 01 2015 *)

Prog

(PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k + x*O(x^n))^-[ 0, 1, 1, 2, 1, 2, 1, 1][k%8 + 1]), n))};

Crossrefs

Sequence in context: A076529 A323283 A279715 * A327047 A332281 A241903

Adjacent sequences: A132209 A132210 A132211 * A132213 A132214 A132215

Keyword

nonn

Author

Michael Somos, Aug 13 2007

Status

approved