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A120006 Expansion of ((eta(q^2) * eta(q^14)) / (eta(q) * eta(q^7)))^3 in powers of q. 3
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 (list; graph; refs; listen; history; text; internal format)
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

M. 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: A058554 A128517 A022568 * A263847 A061567 A293076

Adjacent sequences:  A120003 A120004 A120005 * A120007 A120008 A120009

KEYWORD

nonn

AUTHOR

Michael Somos, Jun 02 2006

STATUS

approved

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Last modified August 13 02:09 EDT 2020. Contains 336441 sequences. (Running on oeis4.)