A193236 Expansion of g.f.: ( Sum_{n>=0} (-3)^n*(2*n+1)*x^(n*(n+1)/2) )^(-1/3).

1, 3, 18, 111, 765, 5481, 40581, 306099, 2342034, 18108270, 141176412, 1108011312, 8744143401, 69325981191, 551800999215, 4406974587918, 35300439813735, 283495238613855, 2281964065354899, 18406084773140820, 148734744069134439

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 = 8.36088614706990698425869281816592400881040301747301994754853463880147456... and c = 0.394251238482436178656471237011147433344315090769699127874860345979... - Vaclav Kotesovec, Oct 22 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 + 111*x^3 + 765*x^4 + 5481*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. A111984, A111983, A193237.

Sequence in context: A134092 A000274 A207321 * A357203 A215047 A367045

Adjacent sequences: A193233 A193234 A193235 * A193237 A193238 A193239

Keyword

nonn

Author

Paul D. Hanna, Jul 18 2011

Status

approved