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

1, 10, 170, 6500, 332050, 19784060, 1296395700, 90616189800, 6637652225250, 503852804991500, 39337349077483420, 3142010167321271000, 255747325678297576100, 21150729618673827139000, 1773152567858996728205000, 150409554094012703302602000, 12890454660664800562838261250

Offset: 0

Author

Shel Kaphan, Mar 09 2023

Keywords

• nonn
• walk

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

• Alois P. Heinz, Table of n, a(n) for n = 0..503
• Jonathan Novak, Pólya's Random Walk Theorem.

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.

Programs

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}]]

Formula

G.f.: 2 - 1/Integral_{t=0..oo} exp(-t)*BesselI(0,2*t*sqrt(x))^5 dt.

INVERTi transform of A169714.