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
a(n) ~ 2^(2*n) * 3^(n+1) / (Pi^2 * n^2). - Vaclav Kotesovec, Apr 30 2024
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-François 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
KEYWORD
nonn
AUTHOR
Simon Plouffe, Oct 30 2011
STATUS
approved