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.

1, 1, 4, 36, 720, 23400, 1123200, 74440800, 6509318400, 725829724800, 100511918784000, 16922530756454400, 3404178048774758400, 806369627582929612800, 222159405758654317363200, 70435689828806256514560000, 25463217531292911649057996800, 10411540182139235537714555289600

Offset

0,3

Links

Table of n, a(n) for n=0..17.

Formula

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

Maple

b:= proc(l) option remember; (n-> `if`(n<2 or max(l[])=0, 1,
      add(`if`(l[i]=0 or 1<abs(l[`if`(i=1, 0, i)-1]-l[i]+1)
                      or 1<abs(l[`if`(i=n, 0, i)+1]-l[i]+1), 0,
        b(subsop(i=l[i]-1, l))), i=1..n)))(nops(l))
    end:
a:= n-> b([2$n]):
seq(a(n), n=0..12);  # Alois P. Heinz, Jan 05 2019

Mathematica

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 *)

Crossrefs

Cf. A227656, A318191.

Sequence in context: A163887 A156630 A289545 * A145565 A360903 A214669

Adjacent sequences: A322779 A322780 A322781 * A322783 A322784 A322785

Keyword

nonn,walk

Author

Woong-Gi Jung, Dec 26 2018

Extensions

More terms from Alois P. Heinz, Dec 30 2018

Status

approved