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A242812 Decimal expansion of the expected number of returns to the origin of a random walk on a 4-d lattice.

%I A242812 #21 Feb 16 2025 08:33:22
%S A242812 1,2,3,9,4,6,7,1,2,1,8,4,8,4,8,1,7,1,2,6,7,8,6,9,7,6,6,4,8,5,9,0,0,0,
%T A242812 7,1,0,1,5,3,2,8,9,0,6,9,1,6,1,7,5,8,6,5,6,9,5,3,4,0,1,8,5,0,7,1,6,2,
%U A242812 8,1,3,3,8,6,5,5,5,6,3,3,3,1,0,3,2,3,9,3,3,0,4,7,3,5,3,8,9,3,9,2,8,5,9,9,1,8
%N A242812 Decimal expansion of the expected number of returns to the origin of a random walk on a 4-d lattice.
%D A242812 Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.9 Polya's random walk constants, p. 323.
%H A242812 Marc Mezzarobba, <a href="/A242812/b242812.txt">Table of n, a(n) for n = 1..10000</a>
%H A242812 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PolyasRandomWalkConstants.html">Pólya's Random Walk Constants</a>.
%F A242812 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 A242812 m(d) = Integral_(t>0) exp(-t)*BesselI(0,t/d)^d dt where BesselI(0,x) is the zeroth modified Bessel function.
%F A242812 Equals 1/(1 - A086232). - _Amiram Eldar_, Aug 28 2020
%e A242812 1.239467121848481712678697664859...
%p A242812 m4:= int(exp(-t)*BesselI(0, t/4)^4, t=0..infinity):
%p A242812 s:= convert(evalf(m4, 120), string):
%p A242812 map(parse, subs("."=NULL, [seq(i, i=s)]))[]; # _Alois P. Heinz_, May 23 2014
%t A242812 digits = 50; NIntegrate[BesselI[0, t/4]^4*Exp[-t], {t, 0, Infinity}, PrecisionGoal -> digits, WorkingPrecision -> 350] // RealDigits [#, 10, digits]& // First (* after _Ryan Propper_ *)
%Y A242812 Cf. A086230, A086231, A086232, A086233, A086234, A086235, A086236, A242813.
%K A242812 nonn,cons
%O A242812 1,2
%A A242812 _Jean-François Alcover_, May 23 2014
%E A242812 More terms from _Alois P. Heinz_, May 23 2014