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A242813 Decimal expansion of the expected number of returns to the origin of a random walk on a 5-d lattice.

%I A242813 #20 Feb 16 2025 08:33:22
%S A242813 1,1,5,6,3,0,8,1,2,4,8,4,0,2,3,1,1,7,8,7,0,7,1,3,5,1,2,1,9,3,8,5,6,6,
%T A242813 9,8,5,5,4,5,4,2,7,3,4,8,5,0,5,1,4,2,3,8,8,2,6,9,5,6,6,0,1,1,2,1,0,0,
%U A242813 8,7,7,0,3,4,7,0,6,8,7,3,1,1,7,2,3,6,6,5,4,3,0,4,9,5,0,9,1,7,1,6,5,2,6,7,4,3
%N A242813 Decimal expansion of the expected number of returns to the origin of a random walk on a 5-d lattice.
%D A242813 Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.9 Polya's random walk constants, p. 323.
%H A242813 Marc Mezzarobba, <a href="/A242813/b242813.txt">Table of n, a(n) for n = 1..10000</a>
%H A242813 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PolyasRandomWalkConstants.html">Pólya's Random Walk Constants</a>.
%F A242813 m(d) = d/(2*Pi)^d*multipleIntegral(-Pi..Pi) (d-sum_(k=1..d) cos(t_k))^(-1) dt_1 dt_2 ... dt_d, where d is the lattice dimension.
%F A242813 m(d) = integral_(t>0) exp(-t)*BesselI(0,t/d)^d dt where BesselI(0,x) is the zeroth modified Bessel function.
%F A242813 Equals 1/(1 - A086233). - _Amiram Eldar_, Aug 28 2020
%e A242813 1.1563081248...
%p A242813 m5:= int(exp(-t)*BesselI(0, t/5)^5, t=0..infinity):
%p A242813 s:= convert(evalf(m5, 120), string):
%p A242813 map(parse, subs("."=NULL, [seq(i, i=s)]))[]; # _Alois P. Heinz_, May 23 2014
%t A242813 d = 5; d/Pi^d*NIntegrate[(d - Sum[Cos[t[k]], {k, 1, d}])^-1, Sequence @@ Table[{t[k], 0, Pi}, {k, 1, d}] // Evaluate] // RealDigits[#, 10, 10]& // First
%o A242813 (PARI) intnumosc(t=0,exp(-t)*besseli(0,t/5)^5,Pi*5) \\ _Charles R Greathouse IV_, Oct 23 2023
%Y A242813 Cf. A086230, A086231, A086232, A086233, A086234, A086235, A086236, A242812, A242814, A242815, A242816.
%K A242813 nonn,cons
%O A242813 1,3
%A A242813 _Jean-François Alcover_, May 23 2014
%E A242813 More terms from _Alois P. Heinz_, May 23 2014