A120006 Expansion of ((eta(q^2) * eta(q^14)) / (eta(q) * eta(q^7)))^3 in powers of q.

1, 3, 6, 13, 24, 42, 73, 123, 201, 320, 504, 774, 1172, 1755, 2592, 3789, 5478, 7851, 11146, 15696, 21942, 30456, 42000, 57546, 78403, 106212, 143124, 191925, 256146, 340320, 450204, 593163, 778416, 1017698, 1325784, 1721157, 2227050, 2872422

Offset

1,2

Comments

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

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q * (chi(-q) * chi(-q^7))^3 in powers of q where chi() is a Ramanujan theta function.

Euler transform of period 14 sequence [ 3, 0, 3, 0, 3, 0, 6, 0, 3, 0, 3, 0, 3, 0, ...].

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 - v - 6*u*v - 8*u*v^2.

G.f.: x * (Product_{k>0} (1 + x^k) * (1 + x^(7*k)))^3.

Convolution inverse of A132319.

a(n) ~ exp(2*Pi*sqrt(2*n/7)) / (8 * 2^(3/4) * 7^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2015

Example

q + 3*q^2 + 6*q^3 + 13*q^4 + 24*q^5 + 42*q^6 + 73*q^7 + 123*q^8 + 201*q^9 + ...

Mathematica

nmax = 40; Rest[CoefficientList[Series[x * Product[((1 + x^k) * (1 + x^(7*k)))^3, {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Sep 07 2015 *)
eta[q_] := q^(1/24)*QPochhammer[q]; Rest[CoefficientList[Series[(( eta[q^2]*eta[q^14])/(eta[q]*eta[q^7]))^3, {q, 0, 50}], q]] (* G. C. Greubel, Apr 19 2018 *)

Prog

(PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^14 + A) / (eta(x + A) * eta(x^7 + A)))^3, n))}

Crossrefs

Cf. A132319.

Sequence in context: A342853 A128517 A022568 * A263847 A061567 A293076

Adjacent sequences: A120003 A120004 A120005 * A120007 A120008 A120009

Keyword

nonn

Author

Michael Somos, Jun 02 2006

Status

approved