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.

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

Offset

1,3

Comments

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

Links

Table of n, a(n) for n=1..60.

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

Cf. A112194.

Sequence in context: A307857 A116921 A173989 * A097357 A342016 A360923

Adjacent sequences: A093065 A093066 A093067 * A093069 A093070 A093071

Keyword

nonn

Author

Michael Somos, Mar 17 2004

Status

approved