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A198808 Number of closed paths of length n whose steps are 12th roots of unity, U_12(n). 5
1, 0, 12, 24, 396, 2160, 23160, 186480, 1845900, 17213280, 171575712, 1703560320, 17365421304, 178323713568, 1856554560432, 19487791106784, 206411964321420, 2201711191213248, 23642813637773616, 255355132936441824, 2772650461148938656 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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.

LINKS

Robert Israel, Table of n, a(n) for n = 0..929

V. Braun, P. Candelas, X. de la Ossa, Two One-Parameter Special Geometries, arXiv preprint arXiv:1512.08367 [hep-th], 2015.

Gilbert Labelle and Annie Lacasse, Closed paths whose steps are roots of unity, in FPSAC 2011, Reykjavik, Iceland DMTCS proc. AO, 2011, 599-610.

FORMULA

E.g.f.: g(x)^2, where g(x) is the e.g.f. of A002898. - Robert Israel, Nov 15 2016

MAPLE

N:= 50: # to get a(0)..a(N)

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):

S:= add(U6(n)*x^n/n!, n=0..N)^2:

seq(coeff(S, x, n)*n!, n=0..N); # Robert Israel, Nov 15 2016

MATHEMATICA

terms = 21;

g[x_] = BesselI[0, 2x]^3 + 2 Sum[BesselI[k, 2x]^3, {k, 1, terms}];

CoefficientList[g[x]^2 + O[x]^terms, x] Range[0, terms-1]! (* Jean-Fran├žois Alcover, Sep 18 2018, after Robert Israel *)

PROG

(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

CROSSREFS

Cf. A002898, A198800.

Sequence in context: A002167 A154268 A058994 * A249134 A323195 A033165

Adjacent sequences:  A198805 A198806 A198807 * A198809 A198810 A198811

KEYWORD

nonn

AUTHOR

Simon Plouffe, Oct 30 2011

STATUS

approved

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Last modified April 19 17:46 EDT 2021. Contains 343117 sequences. (Running on oeis4.)