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A361364 Number of 5-dimensional cubic lattice walks that start and end at origin after 2n steps, not touching origin at intermediate stages.

%I A361364 #24 Nov 15 2024 19:30:11
%S A361364 1,10,170,6500,332050,19784060,1296395700,90616189800,6637652225250,
%T A361364 503852804991500,39337349077483420,3142010167321271000,
%U A361364 255747325678297576100,21150729618673827139000,1773152567858996728205000,150409554094012703302602000,12890454660664800562838261250
%N A361364 Number of 5-dimensional cubic lattice walks that start and end at origin after 2n steps, not touching origin at intermediate stages.
%C A361364 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 5-d case is the series corresponding to A287317.
%C A361364 To satisfy this g.f. equation, a(0) should be 0, but we give it as 1 since there is one trivial loop of 0 steps, and for consistency with related sequences.
%C A361364 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) = 100^n.
%H A361364 Alois P. Heinz, <a href="/A361364/b361364.txt">Table of n, a(n) for n = 0..503</a>
%H A361364 Jonathan Novak, <a href="https://math.mit.edu/classes/18.095/lect2/notes.pdf">Pólya's Random Walk Theorem</a>.
%F A361364 G.f.: 2 - 1/Integral_{t=0..oo} exp(-t)*BesselI(0,2*t*sqrt(x))^5 dt.
%F A361364 INVERTi transform of A169714.
%t A361364 walk5d[n_] :=
%t A361364  Sum[(2 n)!/(i! j! k! l! (n - i - j - k - l)!)^2, {i, 0, n}, {j, 0,
%t A361364    n - i}, {k, 0, n - i - j}, {l, 0, n - i - j - k}]; invertSeq[seq_] :=
%t A361364  CoefficientList[1 - 1/SeriesData[x, 0, seq, 0, Length[seq], 1], x]; invertSeq[Table[walk5d[n], {n, 0, 15}]]
%Y A361364 Cf. A287317, A039699 (number of walks that return to the origin in 2n steps).
%Y A361364 Number of walks that return to the origin for the first time in 2n steps, in 1..4 dimensions: |A002420|, A054474, A049037, A359801.
%Y A361364 Column k=5 of A361397.
%Y A361364 Cf. A169714.
%K A361364 nonn,walk
%O A361364 0,2
%A A361364 _Shel Kaphan_, Mar 09 2023