A261154 Expansion of psi(q^6) * f(-q^12) / (psi(-q) * psi(q^9)) in powers of q where psi(), f() are Ramanujan theta functions.

1, 1, 1, 2, 3, 4, 6, 8, 11, 14, 18, 24, 30, 38, 48, 60, 75, 92, 114, 140, 170, 208, 252, 304, 366, 439, 526, 626, 744, 884, 1044, 1232, 1451, 1704, 1998, 2336, 2730, 3182, 3700, 4300, 4986, 5772, 6672, 7700, 8876, 10212, 11736, 13472, 15438, 17673, 20207

Offset: 0

Author

Michael Somos, Aug 10 2015

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 + ...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..2000
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A186115, A212484, A233693.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ 2^(1/2) q^(1/2) EllipticTheta[ 2, 0, q^3] QPochhammer[ q^12] / (EllipticTheta[ 2, Pi/4, q^(1/2)] EllipticTheta[ 2, 0, q^(9/2)]), {q, 0, n}];

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^9 + A) * eta(x^12 + A)^3 / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A) * eta(x^18 + A)^2), n))};

Formula

Expansion of eta(q^2) * eta(q^9) * eta(q^12)^3 / (eta(q) * eta(q^4) * eta(q^6) * eta(q^18)^2) in powers of q.

Euler transform of period 36 sequence [1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, -1, 1, 0, 1, 1, 1, 2, 1, 1, 1, 0, 1, -1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 1/2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A186115.

a(n) = A233693(n) unless n=0. a(2*n) = A212484(n).

a(n) ~ exp(2*Pi*sqrt(n)/3) / (4*sqrt(3)*n^(3/4)). - Vaclav Kotesovec, Nov 16 2017