OFFSET
-1,2
COMMENTS
Equals the triangular series convolved with the aerated variant of A153084; i.e., (1, 3, 6, 10, ...) * (1, 0, 3, 0, 9, ...). Note that (1, 3, 6, 10, ...) convolved with (1, 0, 3, 0, 6, ...) = A038163: (1, 3, 9, 19, 39, 69, ...). - Gary W. Adamson, Aug 12 2016
LINKS
G. A. Edgar, Table of n, a(n) for n = -1..997
FORMULA
Expansion of (eta(q^3) * eta(q^5) / (eta(q) * eta(q^15)))^3 in powers of q.
Euler transform of period 15 sequence [ 3, 3, 0, 3, 0, 0, 3, 3, 0, 0, 3, 0, 3, 3, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u + v) * (u^2 + 5*u*v + v^2) - u*v * (u*v - 1).
G.f. is a period 1 Fourier series which satisfies f(-1 / (15 t)) = f(t) where q = exp(2 Pi i t).
G.f. x^(-1) * (Prod_{k>0} ( (1 - x^(3*k)) * (1 - x^(5*k)) / ((1 - x^k) * (1 - x^(15*k)))))^3. [corrected by Vaclav Kotesovec, Jun 26 2018]
a(n) ~ exp(4*Pi*sqrt(n/15)) / (sqrt(2) * 15^(1/4) * n^(3/4)). - Vaclav Kotesovec, Jun 26 2018
EXAMPLE
T15C = 1/q + 3 + 9*q + 19*q^2 + 42*q^3 + 78*q^4 + 146*q^5 + 249*q^6 + 429*q^7 + ...
42 = (1, 3, 6, 10, 15) dot (9, 0, 3, 0, 1) = (9 + 0 + 18 + 0 + 15). - Gary W. Adamson, Aug 12 2016
MATHEMATICA
nmax=60; CoefficientList[Series[Product[(((1-x^(3*k)) * (1-x^(5*k)) / (1-x^(15*k)) / (1-x^k)))^3, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 13 2015, typo corrected by Vaclav Kotesovec, Jun 26 2018 *)
PROG
(PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x^3 + A) * eta(x^5 + A) / (eta(x + A) * eta(x^15 + A)))^3, n))}
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Dec 18 2008
STATUS
approved