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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.
2
1, 10, 170, 6500, 332050, 19784060, 1296395700, 90616189800, 6637652225250, 503852804991500, 39337349077483420, 3142010167321271000, 255747325678297576100, 21150729618673827139000, 1773152567858996728205000, 150409554094012703302602000, 12890454660664800562838261250
OFFSET
0,2
COMMENTS
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.
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.
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.
LINKS
FORMULA
G.f.: 2 - 1/Integral_{t=0..oo} exp(-t)*BesselI(0,2*t*sqrt(x))^5 dt.
INVERTi transform of A169714.
MATHEMATICA
walk5d[n_] :=
Sum[(2 n)!/(i! j! k! l! (n - i - j - k - l)!)^2, {i, 0, n}, {j, 0,
n - i}, {k, 0, n - i - j}, {l, 0, n - i - j - k}]; invertSeq[seq_] :=
CoefficientList[1 - 1/SeriesData[x, 0, seq, 0, Length[seq], 1], x]; invertSeq[Table[walk5d[n], {n, 0, 15}]]
CROSSREFS
Cf. A287317, A039699 (number of walks that return to the origin in 2n steps).
Number of walks that return to the origin for the first time in 2n steps, in 1..4 dimensions: |A002420|, A054474, A049037, A359801.
Column k=5 of A361397.
Cf. A169714.
Sequence in context: A034830 A098345 A366299 * A119043 A187204 A346458
KEYWORD
nonn,walk
AUTHOR
Shel Kaphan, Mar 09 2023
STATUS
approved