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
Examples
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, ...
Links
- Alois P. Heinz, Antidiagonals n = 0..140, flattened
Crossrefs
Programs
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Maple
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); -
Mathematica
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 *)
Formula
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
Comments