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A303351 Expansion of Product_{n>=1} (1 + 9*x^n)^(1/3). 2
1, 3, -6, 57, -294, 1884, -13011, 95178, -712293, 5448495, -42444375, 335392941, -2681006280, 21639853488, -176113016241, 1443450932445, -11903668996713, 98695838478585, -822212761531101, 6878755556938029, -57767592614370576, 486792969548157129 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -1/3, g(n) = -9.

In general, if h > 1 and g.f. = Product_{k>=1} (1 + h^2*x^k)^(1/h), then a(n) ~ -(-1)^n * c^(1/h) * h^(2*n-1) / (Gamma(1 - 1/h) * n^(1 + 1/h)), where c = Product_{k>=2} (1 + (-1)^k / h^(2*k-2)). - Vaclav Kotesovec, Apr 22 2018

LINKS

Table of n, a(n) for n=0..21.

FORMULA

a(n) ~ -(-1)^n * c^(1/3) * 3^(2*n-1) / (Gamma(2/3) * n^(4/3)), where c = Product_{k>=2} (1 + 9*(-1/9)^k) = 1.09874828793226302381837574278380702... - Vaclav Kotesovec, Apr 22 2018

MAPLE

seq(coeff(series(mul((1+9*x^k)^(1/3), k = 1..n), x, n+1), x, n), n = 0..25); # Muniru A Asiru, Apr 22 2018

MATHEMATICA

nmax = 30; CoefficientList[Series[Product[(1 + 9*x^k)^(1/3), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 22 2018 *)

PROG

(PARI) N=66; x='x+O('x^N); Vec(prod(k=1, N, (1+9*x^k)^(1/3)))

CROSSREFS

Expansion of Product_{n>=1} (1 + b^2*x^n)^(1/b): A000009 (b=1), A303350 (b=2), this sequence (b=3).

Sequence in context: A051641 A003098 A045914 * A067610 A067609 A012473

Adjacent sequences:  A303348 A303349 A303350 * A303352 A303353 A303354

KEYWORD

sign

AUTHOR

Seiichi Manyama, Apr 22 2018

STATUS

approved

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Last modified November 19 19:18 EST 2019. Contains 329323 sequences. (Running on oeis4.)