A193237 Expansion of g.f.: ( Sum_{n>=0} (2*n+1)*3^n*(-x)^(n*(n+1)/2) )^(-1/3).

1, 3, 18, 141, 1125, 9261, 78255, 673137, 5864238, 51592770, 457484382, 4082618376, 36627119109, 330070717935, 2985857903655, 27099681108948, 246666402397287, 2250904657271427, 20586440729350197, 188659279149885810, 1732045683183434379

Offset

0,2

Comments

Compare to the q-series identity:

eta(x)^3 = Sum_{n>=0} (-1)^n*(2*n+1) * x^(n*(n+1)/2),

where eta(x) is Dedekind's eta(q) function without the q^(1/24) factor.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Formula

a(n) ~ c * d^n / n^(2/3), where d = 9.4965311156864435218580751434880111273815124046288142684565335715382874398... and c = 0.36119424767629376647844389305140717053117960043693037845743425313588325... - Vaclav Kotesovec, Oct 20 2020

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

Example

G.f.: A(x) = 1 + 3*x + 18*x^2 + 141*x^3 + 1125*x^4 + 9261*x^5 +...
where
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) +...

Maple

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

Mathematica

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 *)

Prog

(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)}
(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

Crossrefs

Cf. A111983, A111984, A193236.

Sequence in context: A127129 A186266 A260506 * A325996 A364417 A212030

Adjacent sequences: A193234 A193235 A193236 * A193238 A193239 A193240

Keyword

nonn

Author

Paul D. Hanna, Jul 18 2011

Status

approved