A132968 Expansion of chi(-q) * chi(-q^15) / (chi(-q^6) * chi(-q^10)) in powers of q where chi() is a Ramanujan theta function.

1, -1, 0, -1, 1, -1, 2, -2, 2, -3, 4, -4, 5, -6, 6, -9, 11, -10, 14, -16, 17, -22, 24, -26, 32, -37, 40, -47, 54, -58, 70, -80, 84, -100, 112, -122, 143, -158, 172, -198, 222, -242, 274, -306, 332, -379, 422, -454, 515, -569, 620, -698, 766, -834, 932, -1028

Offset

0,7

Comments

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

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

Formula

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

Euler transform of period 60 sequence [-1, 0, -1, 0, -1, 1, -1, 0, -1, 1, -1, 0, -1, 0, -2, 0, -1, 1, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 2, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 1, -1, 0, -2, 0, -1, 0, -1, 1, -1, 0, -1, 1, -1, 0, -1, 0, -1, 0, ...].

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 A132967.

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

a(n) = - A132967(n) unless n=0.

a(n) ~ (-1)^n * exp(2*Pi*sqrt(n/15)) / (4 * 15^(1/4) * n^(3/4)). - Vaclav Kotesovec, Nov 07 2015

Example

G.f. = 1 - q - q^3 + q^4 - q^5 + 2*q^6 - 2*q^7 + 2*q^8 - 3*q^9 + 4*q^10 - ...

Mathematica

nmax = 60; CoefficientList[Series[Product[(1 + x^(6*k)) * (1 + x^(10*k)) / ( (1 + x^k) * (1 + x^(15*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 07 2015 *)

Prog

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

Crossrefs

Cf. A132967.

Sequence in context: A053254 A067357 A051059 * A132967 A029075 A029052

Adjacent sequences: A132965 A132966 A132967 * A132969 A132970 A132971

Keyword

sign

Author

Michael Somos, Sep 02 2007

Status

approved