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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 `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 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 *) 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 August 5 22:11 EDT 2020. Contains 336214 sequences. (Running on oeis4.)