A193237 - OEIS

%I #17 Feb 27 2021 21:30:02

%S 1,3,18,141,1125,9261,78255,673137,5864238,51592770,457484382,

%T 4082618376,36627119109,330070717935,2985857903655,27099681108948,

%U 246666402397287,2250904657271427,20586440729350197,188659279149885810,1732045683183434379

%N Expansion of g.f.:  ( Sum_{n>=0} (2*n+1)*3^n*(-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="/A193237/b193237.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) ~ c * d^n / n^(2/3), where d = 9.4965311156864435218580751434880111273815124046288142684565335715382874398... and c = 0.36119424767629376647844389305140717053117960043693037845743425313588325... - _Vaclav Kotesovec_, Oct 20 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 + 141*x^3 + 1125*x^4 + 9261*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. A111983, A111984, A193236.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jul 18 2011