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%I #21 Apr 18 2023 09:39:55
%S 1,1,1,1,9,41,121,2241,18481,91729,2577681,30833441,215554681,
%T 8126363961,127462383049,1150296157921,54416525377761,
%U 1056352067669921,11684649751431841,665061201610232769,15390714465319910761,201615391902487799881
%N Expansion of e.g.f. exp( (-LambertW(-x^3))^(1/3) ).
%C 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
%H Seiichi Manyama, <a href="/A362293/b362293.txt">Table of n, a(n) for n = 0..428</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/3)} A362292(k) * binomial(n-1,3*k) * a(n-3*k-1).
%F 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
%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp((-lambertw(-x^3))^(1/3))))
%Y Cf. A000272, A362283, A362292, A362300.
%K nonn
%O 0,5
%A _Seiichi Manyama_, Apr 14 2023
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