login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

A132212
Expansion of f(-x, -x^7) / f(-x, -x) in powers of q where f(, ) is Ramanujan's general theta function.
1
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
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
KEYWORD
nonn
AUTHOR
Michael Somos, Aug 13 2007
STATUS
approved