|
| |
|
|
A132217
|
|
Expansion of psi(x^6) / psi(-x) in powers of x where psi() is a Ramanujan theta function.
|
|
5
|
|
|
|
1, 1, 1, 2, 3, 4, 6, 8, 11, 15, 19, 25, 33, 42, 53, 68, 86, 107, 134, 166, 205, 253, 309, 377, 460, 557, 672, 811, 974, 1166, 1394, 1661, 1975, 2344, 2773, 3275, 3863, 4543, 5333, 6253, 7316, 8544, 9964, 11600, 13484, 15653, 18140, 20994, 24269, 28011, 32288
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,4
|
|
|
COMMENTS
|
|
|
|
LINKS
|
|
|
|
FORMULA
|
Expansion of q^(-5/8) * eta(q^2) * eta(q^12)^2 / (eta(q) * eta(q^4) * eta(q^6)) in powers of q.
Euler transform of period 12 sequence [ 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, ...].
Product_{k>0} (1 - x^(12*k)) * (1 - x^(2*k) + x^(4*k)) / (1 - x^k).
Expansion of f(-x^2, -x^10) / f(-x, -x^2) in powers of x where f(, ) is Ramanujan's general theta function. - Michael Somos, Oct 06 2015
Number of partitions of n into parts not congruent to 0, 2, 10 (mod 12). - Michael Somos, Oct 06 2015
a(n) ~ exp(sqrt(n/2)*Pi) / (2^(11/4) * sqrt(3) * n^(3/4)). - Vaclav Kotesovec, Oct 06 2015
|
|
|
EXAMPLE
|
G.f. = 1 + x + x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 6*x^6 + 8*x^7 + 11*x^8 + 15*x^9 + ...
G.f. = q^5 + q^13 + q^21 + 2*q^29 + 3*q^37 + 4*q^45 + 6*q^53 + 8*q^61 + 11*q^69 + ...
|
|
|
MATHEMATICA
|
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x^3] / (2^(1/2) x^(5/8) EllipticTheta[ 2, Pi/4, x^(1/2)]), {x, 0, n}]; (* Michael Somos, Oct 06 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x^12] QPochhammer[ x^2, x^12] QPochhammer[ x^10, x^12] / QPochhammer[ x], {x, 0, n}]; (* Michael Somos, Oct 06 2015 *)
|
|
|
PROG
|
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff(eta(x^2 + A) * eta(x^12 + A)^2 / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A)), n))};
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
