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

0,4

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^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

Example

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 + ...

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 *)

Prog

(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))};

Crossrefs

Cf. A058728, A094203, A132968, A143752.

Sequence in context: A140952 A096911 A143752 * A300788 A120171 A364915

Adjacent sequences: A145930 A145931 A145932 * A145934 A145935 A145936

Keyword

nonn

Author

Michael Somos, Oct 25 2008

Status

approved