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%I #23 Apr 30 2024 03:24:13
%S 1,0,18,36,918,5400,82800,801360,10907190,132053040,1802041668,
%T 24199809480,340640607384,4834708246368,70229958125184,
%U 1032223723667136,15391538570569590,231935110984687968,3531542904056225916,54244559313713885688,839979883121036697468
%N Number of closed paths of length n whose steps are 18th roots of unity, U_18(n).
%C U_18(n), comment in article: For each m >= 1, the sequence (U_m(N)), N >= 0 is P-recursive but is not algebraic when m > 2.
%H Andrew Howroyd, <a href="/A198802/b198802.txt">Table of n, a(n) for n = 0..200</a>
%H Gilbert Labelle and Annie Lacasse, <a href="https://doi.org/10.46298/dmtcs.2937">Closed paths whose steps are roots of unity</a>, in FPSAC 2011, Reykjavik, Iceland DMTCS proc. AO, 2011, 599-610.
%F E.g.f.: g(x)^3 where g(x) is the e.g.f. of A002898.
%F a(n) ~ 2^(n-3) * 3^(2*n + 9/2) / (Pi^3 * n^3). - _Vaclav Kotesovec_, Apr 30 2024
%o (PARI) seq(n)={Vec(serlaplace(sum(k=0, n, if(k,2,1)*(x^k*besseli(k, 2*x + O(x^(n-k+1)))/k!)^3)^3))} \\ _Andrew Howroyd_, Nov 01 2018
%Y Cf. A002898, A198808.
%K nonn
%O 0,3
%A _Simon Plouffe_, Oct 30 2011