|
|
A128770
|
|
Expansion of phi(-q^9) / phi(-q) in powers of q where phi() is a Ramanujan theta function.
|
|
4
|
|
|
1, 2, 4, 8, 14, 24, 40, 64, 100, 152, 228, 336, 488, 700, 992, 1392, 1934, 2664, 3640, 4936, 6648, 8896, 11832, 15648, 20584, 26942, 35096, 45512, 58768, 75576, 96816, 123568, 157156, 199200, 251676, 316992, 398072, 498460, 622448, 775216, 963012
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,2
|
|
COMMENTS
|
|
|
LINKS
|
|
|
FORMULA
|
Expansion of eta(q^2) * eta(q^9)^2 / ( eta(q)^2 * eta(q^18) ) in powers of q.
Euler transform of period 18 sequence [ 2, 1, 2, 1, 2, 1, 2, 1, 0, 1, 2, 1, 2, 1, 2, 1, 2, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u-1) * (v^2-u) - 2*u*v * (1-v).
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (v-u)^3 - v*(3*u-1) * (1-v) * (1 - 2*v + 3*u*v).
G.f.: Product_{k>0} (1 + x^k) * (1 - x^(9*k)) / ( (1 - x^k) * (1+x^(9*k)) ).
a(n) ~ exp(2*Pi*sqrt(2*n)/3) / (2^(3/4) * 3^(3/2) * n^(3/4)). - Vaclav Kotesovec, Oct 13 2015
|
|
EXAMPLE
|
G.f. = 1 + 2*x + 4*x^2 + 8*x^3 + 14*x^4 + 24*x^5 + 40*x^6 + 64*x^7 + 100*x^8 + ...
|
|
MATHEMATICA
|
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q^9] / EllipticTheta[ 4, 0, q], {q, 0, n}]; (* Michael Somos, Apr 26 2015 *)
nmax=60; CoefficientList[Series[Product[(1+x^k) * (1-x^(9*k)) / ((1-x^k) * (1+x^(9*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 13 2015 *)
|
|
PROG
|
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^9 + A)^2 / (eta(x + A)^2 * eta(x^18 + A)), n))};
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|