0,8

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

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, ...

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);

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-François Alcover, Dec 12 2013, translated from Maple *)

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: A069777 A382858 A225816 * A064992 A187783 A089759

Adjacent sequences: A227652 A227653 A227654 * A227656 A227657 A227658

nonn,tabl

Alois P. Heinz, Jul 19 2013

approved