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A193237 Expansion of g.f.:  ( Sum_{n>=0} (2*n+1)*3^n*(-x)^(n*(n+1)/2) )^(-1/3). 5
1, 3, 18, 141, 1125, 9261, 78255, 673137, 5864238, 51592770, 457484382, 4082618376, 36627119109, 330070717935, 2985857903655, 27099681108948, 246666402397287, 2250904657271427, 20586440729350197, 188659279149885810, 1732045683183434379 (list; graph; refs; listen; history; text; internal format)
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

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 A212030 A259904

Adjacent sequences:  A193234 A193235 A193236 * A193238 A193239 A193240

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jul 18 2011

STATUS

approved

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Last modified August 7 17:38 EDT 2020. Contains 336278 sequences. (Running on oeis4.)