A259529 Expansion of psi(-x^3)^2 / psi(-x) in powers of x where psi() is a Ramanujan theta function.

1, 1, 1, 0, 1, 2, 2, 2, 3, 3, 3, 3, 5, 6, 5, 6, 8, 9, 10, 10, 13, 15, 15, 17, 20, 23, 24, 25, 30, 34, 36, 39, 45, 50, 53, 57, 65, 73, 77, 83, 94, 104, 110, 118, 132, 145, 154, 166, 185, 201, 214, 230, 253, 276, 293, 316, 346, 375, 399, 427, 467, 505, 537, 575

Offset: 0

Author

Michael Somos, Jun 29 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 + x^4 + 2*x^5 + 2*x^6 + 2*x^7 + 3*x^8 + 3*x^9 + ...
G.f. = q^5 + q^13 + q^21 + q^37 + 2*q^45 + 2*q^53 + 2*q^61 + 3*q^69 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..2500
• Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015.
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A259538.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^{ -1, 0, 1, -1, -1, 0, -1, -1, 1, 0, -1, 1}[[Mod[k, 12, 1]]], {k, n}], {x, 0, n}];
a[ n_] := SeriesCoefficient[ (QPochhammer[ x^3, x^6] QPochhammer[ x^12])^2 / ( QPochhammer[ x, x^2] QPochhammer[ x^4]), {x, 0, n}];

PARI

{a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^[ 1, -1, 0, 1, -1, -1, 0, -1, -1, 1, 0, -1][k%12 + 1]), n))};

PARI

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

Formula

Expansion of f(x, x^5)^2 / f(x) in powers of x where f(,) is the general Ramanujan theta function.

Expansion of q^(-5/8) * eta(q^2) * eta(q^3)^2 * eta(q^12)^2 / (eta(q) * eta(q^4) * eta(q^6)^2) in powers of q.

Euler transform of period 12 sequence [ 1, 0, -1, 1, 1, 0, 1, 1, -1, 0, 1, -1, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (768 t)) = (16/3)^1/2 (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A259538.

G.f.: Product_{k>0} (1 + x^k + x^(2*k)) * (1 - x^(3*k)) * (1 - x^(2*k) + x^(4*k)) * (1 + x^(6*k)).

a(n) ~ exp(Pi*sqrt(n/6)) / (6*sqrt(n)). - Vaclav Kotesovec, Jul 11 2016