cp's OEIS Frontend

A198808 Number of closed paths of length n whose steps are 12th roots of unity, U_12(n).

%I A198808 #29 Apr 30 2024 03:55:15
%S A198808 1,0,12,24,396,2160,23160,186480,1845900,17213280,171575712,
%T A198808 1703560320,17365421304,178323713568,1856554560432,19487791106784,
%U A198808 206411964321420,2201711191213248,23642813637773616,255355132936441824,2772650461148938656
%N A198808 Number of closed paths of length n whose steps are 12th roots of unity, U_12(n).
%C A198808 U_12(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 A198808 Robert Israel, <a href="/A198808/b198808.txt">Table of n, a(n) for n = 0..929</a>
%H A198808 V. Braun, P. Candelas, X. de la Ossa, <a href="https://arxiv.org/abs/1512.08367">Two One-Parameter Special Geometries</a>, arXiv preprint arXiv:1512.08367 [hep-th], 2015.
%H A198808 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 A198808 E.g.f.: g(x)^2, where g(x) is the e.g.f. of A002898. - _Robert Israel_, Nov 15 2016
%F A198808 a(n) ~ 2^(2*n) * 3^(n+1) / (Pi^2 * n^2). - _Vaclav Kotesovec_, Apr 30 2024
%p A198808 N:= 50: # to get a(0)..a(N)
%p A198808 U6:= rectoproc({(36*n^2+180*n+216)*a6(n+1)+(24*n^2+144*n+216)*a6(n+2)+(n^2+7*n+12)*a6(n+3)+(-n^2-8*n-16)*a6(n+4), a6(0) = 1, a6(1) = 0, a6(2) = 6, a6(3) = 12}, a6(n),remember):
%p A198808 S:= add(U6(n)*x^n/n!,n=0..N)^2:
%p A198808 seq(coeff(S,x,n)*n!,n=0..N); # _Robert Israel_, Nov 15 2016
%t A198808 terms = 21;
%t A198808 g[x_] = BesselI[0, 2x]^3 + 2 Sum[BesselI[k, 2x]^3, {k, 1, terms}];
%t A198808 CoefficientList[g[x]^2 + O[x]^terms, x] Range[0, terms-1]! (* _Jean-François Alcover_, Sep 18 2018, after _Robert Israel_ *)
%o A198808 (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)^2))} \\ _Andrew Howroyd_, Nov 01 2018
%Y A198808 Cf. A002898, A198800.
%K A198808 nonn
%O A198808 0,3
%A A198808 _Simon Plouffe_, Oct 30 2011