A362283 -id:A362283 - cp's OEIS frontend

A362293 Expansion of e.g.f. exp( (-LambertW(-x^3))^(1/3) ).

1, 1, 1, 1, 9, 41, 121, 2241, 18481, 91729, 2577681, 30833441, 215554681, 8126363961, 127462383049, 1150296157921, 54416525377761, 1056352067669921, 11684649751431841, 665061201610232769, 15390714465319910761, 201615391902487799881

Offset: 0

Seiichi Manyama, Apr 14 2023

nonn

Let k be a positive integer. It appears that reducing this sequence modulo k produces an eventually periodic sequence with period a multiple of k. For example, modulo 3 the sequence becomes [1, 1, 1, 0, 2, 1, 0, 1, 1, 0, 2, 1, 0, 1, 1, 0, 2, 1, 0, ...], with an apparent period [1, 1, 0, 2, 1, 0] of length 6 starting at a(1). - Peter Bala, Apr 16 2023

Seiichi Manyama, Table of n, a(n) for n = 0..428
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A000272, A362283, A362292, A362300.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp((-lambertw(-x^3))^(1/3))))

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/3)} A362292(k) * binomial(n-1,3*k) * a(n-3*k-1).

a(n) ~ (1 + 2*cos(2*Pi*mod(n-1,3)/3 - sqrt(3)/2)/exp(3/2)) * n^(n-1) / (sqrt(3) * exp(2*n/3 - 1)). - Vaclav Kotesovec, Apr 18 2023

cp's OEIS Frontend

A362293 Expansion of e.g.f. exp( (-LambertW(-x^3))^(1/3) ).

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