A145933 Expansion of chi(-q^6) * chi(-q^10) / (chi(-q) * chi(-q^15)) in powers of q where chi() is a Ramanujan theta function.

1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 8, 10, 11, 14, 17, 20, 23, 27, 31, 36, 41, 48, 55, 63, 72, 82, 94, 106, 122, 137, 156, 175, 197, 222, 249, 280, 314, 352, 393, 439, 490, 546, 608, 676, 751, 834, 923, 1024, 1133, 1253, 1384, 1528, 1686, 1857, 2045, 2250, 2474, 2718, 2984

Offset: 0

Author

Michael Somos, Oct 25 2008

Keywords

• nonn

Comments

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

Examples

			G.f. = 1 + q + q^2 + 2*q^3 + 2*q^4 + 3*q^5 + 3*q^6 + 4*q^7 + 5*q^8 + 6*q^9 + ...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000
• Michael Somos, Introduction to Ramanujan theta functions
• Michael Somos, A Remarkable eta-product Identity
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A058728, A094203, A132968, A143752.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x] QPochhammer[ -x^15, x^15] / (QPochhammer[ -x^6, x^6] QPochhammer[ -x^10, x^10]), {x, 0, n}]; (* Michael Somos, Sep 03 2015 *)
nmax = 60; CoefficientList[Series[Product[(1 + x^k) * (1 + x^(15*k)) / ((1 + x^(6*k)) * (1 + x^(10*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 08 2015 *)

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^6 + A) * eta(x^10 + A) * eta(x^30 + A) / (eta(x + A) * eta(x^12 + A) * eta(x^15 + A) * eta(x^20 + A)), n))};

Formula

Expansion of eta(q^2) * eta(q^6) * eta(q^10) * eta(q^30) / (eta(q) * eta(q^12) * eta(q^15) * eta(q^20)) in powers of q.

Euler transform of a period 60 sequence.

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = 2 * (1 - u)^2 * (1 - v)^2 + 2 * u*v * (u - v)^2 + u*v * (v - u^2) * (u - v^2).

G.f. is a period 1 Fourier series which satisfies f(-1 / (60 t)) = g(t) where q = exp(2 Pi i t) and g() is the g.f. for A058728.

G.f.: Product_{k>0} (1 + x^k) * (1 + x^(15*k)) / ((1 + x^(6*k)) * (1 + x^(10*k))).

a(n) = A143752(n) unless n=0. a(2*n) = A094203(n).

Convolution inverse of A132968.

a(n) ~ exp(2*Pi*sqrt(n/15)) / (2 * 15^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 08 2015