A359801 -id:A359801 - cp's OEIS frontend

A361397 Number A(n,k) of k-dimensional cubic lattice walks with 2n steps from origin to origin and avoiding early returns to the origin; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 0, 1, 2, 0, 1, 4, 2, 0, 1, 6, 20, 4, 0, 1, 8, 54, 176, 10, 0, 1, 10, 104, 996, 1876, 28, 0, 1, 12, 170, 2944, 22734, 22064, 84, 0, 1, 14, 252, 6500, 108136, 577692, 275568, 264, 0, 1, 16, 350, 12144, 332050, 4525888, 15680628, 3584064, 858, 0

Offset: 0

Alois P. Heinz, Mar 10 2023

Column k is INVERTi transform of k-th row of A287318.

			Square array A(n,k) begins:
  1,  1,     1,      1,       1,        1,        1, ...
  0,  2,     4,      6,       8,       10,       12, ...
  0,  2,    20,     54,     104,      170,      252, ...
  0,  4,   176,    996,    2944,     6500,    12144, ...
  0, 10,  1876,  22734,  108136,   332050,   796860, ...
  0, 28, 22064, 577692, 4525888, 19784060, 62039088, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=0-5 give: A000007, |A002420|, A054474, A049037, A359801, A361364.
Rows n=0-2 give: A000012, A005843, A139271.
Main diagonal gives A361297.
Cf. A000108, A287318.

b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
      add(b(n-j, i-1)*binomial(n, j)^2, j=0..n))
    end:
g:= proc(n, k) option remember; `if` (n<1, -1,
      -add(g(n-i, k)*(2*i)!*b(i, k)/i!^2, i=1..n))
    end:
A:= (n,k)-> `if`(n=0, 1, `if`(k=0, 0, g(n, k))):
seq(seq(A(n, d-n), n=0..d), d=0..10);

b[n_, 0] = 0; b[n_, 1] = 1; b[0, k_] = 1;
b[n_, k_] := b[n, k] = Sum[Binomial[n, i]^2*b[i, k - 1], {i, 0, n}]; (* A287316 *)
g[n_, k_] := g[n, k] = b[n, k]*Binomial[2 n, n]; (* A287318 *)
a[n_, k_] := a[n, k] = g[n, k] - Sum[a[i, k]*g[n - i, k], {i, 1, n - 1}];
TableForm[Table[a[n, k], {k, 0, 10}, {n, 0, 10}]] (* Shel Kaphan, Mar 14 2023 *)

A(n,1)/2 = A000108(n-1) for n >= 1.

G.f. of column k: 2 - 1/Integral_{t=0..oo} exp(-t)*BesselI(0,2*t*sqrt(x))^k dt. - Shel Kaphan, Mar 19 2023

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

Original entry on oeis.org

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

Offset: 0

Shel Kaphan, Mar 09 2023

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.

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

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.

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}]]

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

INVERTi transform of A169714.

cp's OEIS Frontend

A361397 Number A(n,k) of k-dimensional cubic lattice walks with 2n steps from origin to origin and avoiding early returns to the origin; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula