OFFSET
1,2
COMMENTS
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
KEYWORD
nonn
AUTHOR
Michael Somos, Jun 02 2006
STATUS
approved