%I #13 May 10 2018 23:06:51

%S 1,1,2,2,4,5,7,9,13,16,22,28,37,46,59,74,94,115,144,176,218,265,326,

%T 393,479,574,695,830,996,1184,1414,1673,1988,2344,2770,3254,3828,4482,

%U 5252,6126,7153,8318,9678,11222,13018,15050,17405,20068,23145,26621

%N Expansion of eta(q^3) * eta(q^33) / ( eta(q)* eta(q^11)) in powers of q.

%H G. C. Greubel, <a href="/A128663/b128663.txt">Table of n, a(n) for n = 1..1000</a>

%F Euler transform of period 33 sequence [ 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 2, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 2, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, ...].

%F G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u*v * (1 + 3 * u*v) - (u+v) * (u^2 - 3 * u*v + v^2).

%F G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = v - u^3 + 3 * u*v * (2*u + (1+v) * (1 + 3*u*v)).

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (33 t)) = 1/3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A226009.

%F G.f.: x * Product_{k>0} (1 - x^(3*k)) * (1 - x^(33*k)) / ( (1 - x^k) * (1 - x^(11*k))).

%F Convolution inverse of A226009.

%F a(n) ~ exp(4*Pi*sqrt(n/33)) / (sqrt(2) * 3^(5/4) * 11^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Sep 08 2015

%e q + q^2 + 2*q^3 + 2*q^4 + 4*q^5 + 5*q^6 + 7*q^7 + 9*q^8 + 13*q^9 + ...

%t a[ n_] := SeriesCoefficient[ q QPochhammer[ q^3] QPochhammer[ q^33] / (QPochhammer[ q] QPochhammer[ q^11]), {q, 0, n}]

%t nmax = 40; Rest[CoefficientList[Series[x * Product[(1 - x^(3*k)) * (1 - x^(33*k)) / ( (1 - x^k) * (1 - x^(11*k))), {k, 1, nmax}], {x, 0, nmax}], x]] (* _Vaclav Kotesovec_, Sep 08 2015 *)

%o (PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^3 + A) * eta(x^33 + A) / (eta(x + A) * eta(x^11 + A)), n))}

%Y Cf. A226009.

%K nonn

%O 1,3

%A _Michael Somos_, Mar 19 2007