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%I #19 Feb 27 2021 21:29:55
%S 1,3,18,111,765,5481,40581,306099,2342034,18108270,141176412,
%T 1108011312,8744143401,69325981191,551800999215,4406974587918,
%U 35300439813735,283495238613855,2281964065354899,18406084773140820,148734744069134439
%N Expansion of g.f.: ( Sum_{n>=0} (-3)^n*(2*n+1)*x^(n*(n+1)/2) )^(-1/3).
%C Compare to the q-series identity:
%C eta(x)^3 = Sum_{n>=0} (-1)^n*(2*n+1) * x^(n*(n+1)/2),
%C where eta(x) is Dedekind's eta(q) function without the q^(1/24) factor.
%H G. C. Greubel, <a href="/A193236/b193236.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) ~ c * d^n / n^(2/3), where d = 8.36088614706990698425869281816592400881040301747301994754853463880147456... and c = 0.394251238482436178656471237011147433344315090769699127874860345979... - _Vaclav Kotesovec_, Oct 22 2020
%F Conjectures: a(5*n+4) == 0 (mod 5) and a(7*n+5) == 0 (mod 7) (both checked up to n = 200). - _Peter Bala_, Feb 26 2021
%e G.f.: A(x) = 1 + 3*x + 18*x^2 + 111*x^3 + 765*x^4 + 5481*x^5 +...
%e where
%e 1/A(x)^3 = 1 - 9*x + 45*x^3 - 189*x^6 + 729*x^10 - 2673*x^15 + 9477*x^21 - 32805*x^28 +...+ (-3)^n*(2*n+1)*x^(n*(n+1)/2) +...
%p seq(coeff(series(add((2*n+1)*(-3)^n*x^(n*(n+1)/2), n = 0..40)^(-1/3), x, n+1), x, n), n = 0..30); # _G. C. Greubel_, Nov 05 2019
%t CoefficientList[Series[(Sum[(2n+1)*(-3)^n*x^(n(n+1)/2), {n,0,40}] )^(-1/3), {x,0,30}], x] (* _G. C. Greubel_, Nov 05 2019 *)
%o (PARI) {a(n)=local(S=sum(m=0,sqrtint(2*n),(-3)^m*(2*m+1)*x^(m*(m+1)/2))+x*O(x^n));polcoeff(S^(-1/3),n)}
%o (Sage) [( (sum((2*n+1)*(-3)^n*x^(n*(n+1)/2) for n in (0..40)) )^(-1/3) ).series(x,n+1).list()[n] for n in (0..30)] # _G. C. Greubel_, Nov 05 2019
%Y Cf. A111984, A111983, A193237.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jul 18 2011
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