A227656 -id:A227656 - cp's OEIS frontend

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.

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

Alois P. Heinz, Jul 19 2013

			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, ...

Alois P. Heinz, Antidiagonals n = 0..18, flattened

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.
Main diagonal gives A227673.
Cf. A262809, A263159, A318191.

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 *)

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

Woong-Gi Jung, Dec 26 2018

Cf. A227656, A318191.

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 *)

a(n) = n * A318191(2,n) for n > 0. - Alois P. Heinz, Jan 09 2019

More terms from Alois P. Heinz, Dec 30 2018

cp's OEIS Frontend

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.

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