|
| |
|
|
A123630
|
|
Expansion of q * (chi(-q^3) * chi(-q^5)) / (chi(-q) * chi(-q^15))^2 in powers of q where chi() is a Ramanujan theta function.
|
|
5
|
|
|
|
1, 2, 3, 5, 7, 10, 14, 20, 27, 36, 48, 63, 82, 106, 137, 175, 222, 280, 352, 439, 546, 676, 834, 1024, 1253, 1528, 1857, 2250, 2718, 3276, 3936, 4718, 5640, 6728, 8006, 9507, 11266, 13324, 15726, 18526, 21786, 25574, 29970, 35064, 40961, 47774, 55638
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,2
|
|
|
COMMENTS
|
|
|
|
LINKS
|
|
|
|
FORMULA
|
Euler transform of period 30 sequence [ 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 -v - u*v * (4 + 2*v).
G.f. is a period 1 Fourier series which satisfies f(-1 / (30 t)) = (1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A133098.
G.f.: x * Product_{k>0} (1 + x^k) * (1 + x^(15*k)) * P(30, x^k) where P(n, x) is n th cyclotomic polynomial.
a(n) ~ exp(2*Pi*sqrt(2*n/15)) / (2^(7/4) * 15^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2017
|
|
|
EXAMPLE
|
G.f. = q + 2*q^2 + 3*q^3 + 5*q^4 + 7*q^5 + 10*q^6 + 14*q^7 + 20*q^8 + 27*q^9 + ...
|
|
|
MATHEMATICA
|
f[q_] := QPochhammer[-q, q^2]; A123630[n_] := SeriesCoefficient[q*(f[-q^3]*f[-q^5])/(f[-q]*f[-q^15])^2, {q, 0, n}]; Table[A123630[n], {n, 0, 50}] (* G. C. Greubel, Oct 17 2017 *)
|
|
|
PROG
|
(PARI) {a(n) = local(A); if( n<1, 0, n--; A = x*O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^3 + A) * eta(x^5 + A) * eta(x^30 + A)^2 / (eta(x + A)^2 * eta(x^6 + A) * eta(x^10 + A) * eta(x^15 + A)^2), n))};
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
