A227655
Number A(n,k) of lattice paths from {n}^k to {0}^k using steps that decrement one component by 1 such that for each point (p_1,p_2,...,p_k) we have abs(p_{i}-p_{i+1}) <= 1; square array A(n,k), n>=0, k>=0, read by antidiagonals.
Original entry on oeis.org
1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 6, 4, 1, 1, 1, 24, 44, 8, 1, 1, 1, 120, 896, 320, 16, 1, 1, 1, 720, 29392, 33904, 2328, 32, 1, 1, 1, 5040, 1413792, 7453320, 1281696, 16936, 64, 1, 1, 1, 40320, 93770800, 2940381648, 1897242448, 48447504, 123208, 128, 1, 1
Offset: 0
A(2,2) = 2^2 = 4:
(1,2) (0,1)
/ \ / \
(2,2) (1,1) (0,0)
\ / \ /
(2,1) (1,0)
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, ...
1, 1, 2, 6, 24, 120, ...
1, 1, 4, 44, 896, 29392, ...
1, 1, 8, 320, 33904, 7453320, ...
1, 1, 16, 2328, 1281696, 1897242448, ...
1, 1, 32, 16936, 48447504, 482913033152, ...
Columns k=0+1, 2-10 give:
A000012,
A000079,
A227665,
A227666,
A227667,
A227668,
A227669,
A227670,
A227671,
A227672.
Rows n=0-10 give:
A000012,
A000142,
A227656,
A227657,
A227658,
A227659,
A227660,
A227661,
A227662,
A227663,
A227664.
-
b:= proc(l) option remember; `if`({l[]}={0}, 1, add(
`if`(l[i]=0 or i>1 and 1 `if`(k<2, 1, b([n$k])):
seq(seq(A(n, d-n), n=0..d), d=0..10);
-
b[l_] := b[l] = If[Union[l] == {0}, 1, Sum[If[l[[i]] == 0 || i>1 && 1 < Abs[l[[i-1]] - l[[i]] + 1] || i l[[i]]-1]]], {i, 1, Length[l]}]]; a[n_, k_] := If[k<2, 1, b[Array[n&, k]]]; Table[Table[a[n, d - n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Dec 12 2013, translated from Maple *)
A320443
Number of lattice paths from {n}^n to {0}^n using steps that decrement one component by 1 such that for each point p we have abs(p_{i}-p_{(i mod n)+1}) <= 1 and the first component used is p_1.
Original entry on oeis.org
1, 1, 2, 72, 162000, 34907788800, 1178106009360998400, 8852509935316882311338419200, 20266951127950378632425380239663945561600, 18451222767137832036294073367248809725255450206074566400, 8461986175729341705763214274355536481199826590348664142270203923137228800
Offset: 0
A322782
Number of lattice paths from {2}^n to {0}^n using steps that decrement one component by 1 such that for each point (p_1,p_2,...,p_n) we have abs(p_{i}-p_{i+1}) <= 1 and abs(p_{1}-p_{n}) <= 1.
Original entry on oeis.org
1, 1, 4, 36, 720, 23400, 1123200, 74440800, 6509318400, 725829724800, 100511918784000, 16922530756454400, 3404178048774758400, 806369627582929612800, 222159405758654317363200, 70435689828806256514560000, 25463217531292911649057996800, 10411540182139235537714555289600
Offset: 0
-
b:= proc(l) option remember; (n-> `if`(n<2 or max(l[])=0, 1,
add(`if`(l[i]=0 or 1 b([2$n]):
seq(a(n), n=0..12); # Alois P. Heinz, Jan 05 2019
-
b[l_] := b[l] = With[{n = Length[l]}, If[n < 2 || Max[l ] == 0, 1, Sum[If[ l[[i]] == 0 || 1 < Abs[l[[If[i == 1, 0, i] - 1]] - l[[i]] + 1] || 1 < Abs[l[[If[i == n, 0, i] + 1]] - l[[i]] + 1], 0, b[ReplacePart[l, i -> l[[i]] - 1]]], {i, n}]]];
a[n_] := b[Table[2, {n}]];
a /@ Range[0, 12] (* Jean-François Alcover, May 13 2020, after Alois P. Heinz *)
Showing 1-3 of 3 results.