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A261520 Expansion of Product_{k>=1} ((1+x^k)/(1-x^k))^(3^k). 8
1, 6, 36, 200, 1038, 5160, 24776, 115632, 527172, 2355998, 10349448, 44783064, 191211512, 806737800, 3367294320, 13918479872, 57020736942, 231697484304, 934399998412, 3742041461976, 14888854356840, 58881590423856, 231542984619720, 905666813058384 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Convolution of A144067 and A256142.

In general, for m > 1, if g.f. = Product_{k>=1} ((1+x^k)/(1-x^k))^(m^k), then a(n) ~ m^n * exp(2*sqrt(2*n) - 1 + c) / (sqrt(Pi) * 2^(3/4) * n^(3/4)), where c = 2 * Sum_{j>=1} 1/((2*j+1)*(m^(2*j)-1)).

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Vaclav Kotesovec, Asymptotics of the Euler transform of Fibonacci numbers, arXiv:1508.01796 [math.CO]

Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 27.

FORMULA

a(n) ~ 3^n * exp(2*sqrt(2*n) - 1 + c) / (sqrt(Pi) * 2^(3/4) * n^(3/4)), where c = 2 * Sum_{j>=1} 1/((2*j+1)*(3^(2*j)-1)) = 0.0887630729103166089354170592729856346...

MATHEMATICA

nmax = 40; CoefficientList[Series[Product[((1 + x^k)/(1 - x^k))^(3^k), {k, 1, nmax}], {x, 0, nmax}], x]

CROSSREFS

Cf. A156616, A261519, A260916, A001934, A015128.

Sequence in context: A048980 A200782 A055299 * A232138 A000551 A038157

Adjacent sequences:  A261517 A261518 A261519 * A261521 A261522 A261523

KEYWORD

nonn

AUTHOR

Vaclav Kotesovec, Aug 23 2015

STATUS

approved

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Last modified June 4 07:59 EDT 2020. Contains 334822 sequences. (Running on oeis4.)