|
| |
|
|
A093068
|
|
Expansion of (eta(q^3)^2 * eta(q^7) * eta(q^63)) / (eta(q) * eta(q^9) * eta(q^21)^2) in powers of q.
|
|
0
|
|
|
|
1, 1, 2, 1, 3, 3, 4, 3, 5, 6, 9, 8, 11, 12, 15, 16, 18, 21, 26, 28, 35, 39, 47, 51, 58, 66, 77, 85, 97, 108, 125, 141, 156, 174, 195, 218, 245, 270, 304, 336, 377, 417, 467, 512, 573, 627, 702, 770, 853, 935, 1035, 1136, 1257, 1371, 1515, 1654, 1822, 1989, 2184, 2382
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,3
|
|
|
COMMENTS
|
|
|
|
LINKS
|
|
|
|
FORMULA
|
Expansion of ( (c(q) * b(q^3) * b(q^7) * c(q^21)) / (b(q) * c(q^3) * c(q^7) * b(q^21)) )^(1/4) in powers of q where b(), c() are cubic AGM theta functions.
Euler transform of a period 63 sequence.
G.f. A(x) = B(x) / B(x^7) where B() is the g.f. of A112194.
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) = f(1/A(x), 1/A(x^2)) where f(u, v) = u^3 + v^3 - 2*u*v*(u + v) - u^2*v^2 - u*v.
G.f.: x * Product_{k>0} (1 - x^(3*k))^2 * (1 - x^(7*k)) * (1 - x^(63*k)) / ( (1 - x^k) * (1 - x^(9*k)) * (1 - x^(21*k))^2 ).
a(n) ~ exp(4*Pi*sqrt(n/7)/3) / (sqrt(6) * 7^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 08 2015
|
|
|
EXAMPLE
|
G.f. = q + q^2 + 2*q^3 + q^4 + 3*q^5 + 3*q^6 + 4*q^7 + 3*q^8 + 5*q^9 + 6*q^10 + 9*q^11 + ...
|
|
|
MATHEMATICA
|
a[ n_] := SeriesCoefficient[ q QPochhammer[ q^3]^2 QPochhammer[ q^7] QPochhammer[ q^63] / (QPochhammer[ q] QPochhammer[ q^9] QPochhammer[ q^21]^2), {q, 0, n}]; (* Michael Somos, Apr 24 2015 *)
nmax = 60; Rest[CoefficientList[Series[x * Product[(1 - x^(3*k))^2 * (1 - x^(7*k)) * (1 - x^(63*k)) / ( (1 - x^k) * (1 - x^(9*k)) * (1 - x^(21*k))^2 ), {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Sep 08 2015 *)
|
|
|
PROG
|
(PARI) {a(n) = my(A, u, v); if( n<0, 0, A = x; for( k=2, n, u = A + x * O(x^k); v = subst(u, x, x^2); A -= x^k * polcoeff( u^3 + v^3 - 2*u*v*(u + v) - u^2*v^2 - u*v, k+2) / 2); polcoeff(A, n))};
(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^3 + A)^2 * eta(x^7 + A) * eta(x^63 + A) / (eta(x + A) * eta(x^9 + A) * eta(x^21 + A)^2), n))};
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
