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A359801 Number of 4-dimensional cubic lattice walks that start and end at origin after 2n steps, not touching origin at intermediate stages.

%I A359801 #52 Nov 15 2024 19:18:02
%S A359801 1,8,104,2944,108136,4525888,204981888,9792786432,486323201640,
%T A359801 24874892400064,1302278744460352,69474942954714112,
%U A359801 3764568243058030208,206675027529594291200,11473858525271117889536,643154944963894079717376,36355546411928157876528744,2070313613815122857027563200
%N A359801 Number of 4-dimensional cubic lattice walks that start and end at origin after 2n steps, not touching origin at intermediate stages.
%C A359801 In Novak's note it is mentioned that if P(z) and Q(z) are the g.f.s for the probabilities of indecomposable and decomposable loops, respectively, that P(z) = 1 - 1/Q(z). This works equally well using loop counts rather than probabilities. The g.f.s may be expressed by the series constructed from the sequences of counts of loops of length 2*n. Q(z) for the 4-d case is the series corresponding to A039699.
%C A359801 To obtain the probability of returning to the point of origin for the first time after 2*n steps, divide a(n) by the total number of walks of length 2*n in d dimensions: (2*d)^(2*n) = 64^n.
%H A359801 Vaclav Kotesovec, <a href="/A359801/b359801.txt">Table of n, a(n) for n = 0..550</a>
%H A359801 Dorin Dumitraşcu and Liviu Suciu, <a href="https://arxiv.org/abs/2212.01702">Asymptotics for the Number of Random Walks in the Euclidean Lattice</a>, arXiv:2212.01702 [math.CO], 2022, p.11.
%H A359801 Jonathan Novak, <a href="https://math.mit.edu/classes/18.095/lect2/notes.pdf">Pólya's Random Walk Theorem</a>, The American Mathematical Monthly, Vol. 121, No. 8 (October 2014), pp. 711-716.
%F A359801 G.f.: 2 - 1/Q(x) where Q(x) is the g.f. of A039699.
%F A359801 G.f.: 2 - 1/Integral_{t=0..oo} exp(-t)*BesselI(0,2*t*sqrt(x))^4 dt.
%F A359801 INVERTi transform of A039699.
%t A359801 walk4d[n_] :=
%t A359801  Sum[(2 n)!/(i! j! k! (n - i - j - k)!)^2, {i, 0, n}, {j, 0,
%t A359801    n - i}, {k, 0, n - i - j}]; invertSeq[seq_] :=
%t A359801   CoefficientList[1 - 1/SeriesData[x, 0, seq, 0, Length[seq], 1], x]; invertSeq[Table[walk4d[n], {n, 0, 17}]]
%o A359801 (PARI) seq(n) = {my(v=Vec(2 - 1/serlaplace(besseli(0, 2*x + O(x^(2*n+1)))^4))); vector(n+1, i, v[2*i-1])} \\ _Andrew Howroyd_, Mar 08 2023
%Y A359801 Cf. A039699, A287317 (number of walks that return to the origin in 2n steps).
%Y A359801 Number of walks that return to the origin for the first time in 2n steps, in 1..3 dimensions: |A002420|, A054474, A049037.
%Y A359801 Column k=4 of A361397.
%K A359801 nonn,walk
%O A359801 0,2
%A A359801 _Shel Kaphan_, Mar 08 2023