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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. 24
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

LINKS

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

EXAMPLE

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, ...

MAPLE

b:= proc(l) option remember; `if`({l[]}={0}, 1, add(

      `if`(l[i]=0 or i>1 and 1<abs(l[i-1]-l[i]+1) or

       i<nops(l) and 1<abs(l[i+1]-l[i]+1), 0,

       b(subsop(i=l[i]-1, l))), i=1..nops(l)))

    end:

A:= (n, k)-> `if`(k<2, 1, b([n$k])):

seq(seq(A(n, d-n), n=0..d), d=0..10);

MATHEMATICA

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<Length[l] && 1<Abs[l[[i+1]] - l[[i]] + 1], 0, b[ReplacePart[l, 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-Fran├žois Alcover, Dec 12 2013, translated from Maple *)

CROSSREFS

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.

Sequence in context: A213275 A069777 A225816 * A064992 A187783 A089759

Adjacent sequences:  A227652 A227653 A227654 * A227656 A227657 A227658

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Jul 19 2013

STATUS

approved

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Last modified September 20 08:13 EDT 2019. Contains 327214 sequences. (Running on oeis4.)