A132217 Expansion of psi(x^6) / psi(-x) in powers of x where psi() is a Ramanujan theta function.

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

Offset: 0

Author

Michael Somos, Aug 13 2007

Keywords

• nonn

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			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 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A262987.

Programs

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 *)

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))};

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(2*n) = A262987(n). - 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