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A214246
Number A(n,k) of compositions of n where differences between neighboring parts are in {-k,0,k}; square array A(n,k), n>=0, k>=0, read by antidiagonals.
10
1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 4, 3, 1, 1, 2, 2, 6, 2, 1, 1, 2, 2, 5, 11, 4, 1, 1, 2, 2, 3, 5, 17, 2, 1, 1, 2, 2, 3, 4, 10, 29, 4, 1, 1, 2, 2, 3, 2, 7, 10, 47, 3, 1, 1, 2, 2, 3, 2, 6, 8, 21, 78, 4, 1, 1, 2, 2, 3, 2, 4, 5, 9, 22, 130, 2
OFFSET
0,6
LINKS
EXAMPLE
A(3,0) = 2: [3], [1,1,1].
A(4,1) = 6: [4], [2,2], [2,1,1], [1,2,1], [1,1,2], [1,1,1,1].
A(5,2) = 5: [5], [3,1,1], [1,3,1], [1,1,3], [1,1,1,1,1].
A(6,3) = 7: [6], [4,1,1], [3,3], [2,2,2], [1,4,1], [1,1,4], [1,1,1,1,1,1].
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, 1, 1, ...
2, 2, 2, 2, 2, 2, 2, 2, ...
2, 4, 2, 2, 2, 2, 2, 2, ...
3, 6, 5, 3, 3, 3, 3, 3, ...
2, 11, 5, 4, 2, 2, 2, 2, ...
4, 17, 10, 7, 6, 4, 4, 4, ...
2, 29, 10, 8, 5, 4, 2, 2, ...
MAPLE
b:= proc(n, i, k) option remember;
`if`(n<1 or i<1, 0, `if`(n=i, 1, add(b(n-i, i+j, k), j={-k, 0, k})))
end:
A:= (n, k)-> `if`(n=0, 1, add(b(n, j, k), j=1..n)):
seq(seq(A(n, d-n), n=0..d), d=0..15);
MATHEMATICA
b[n_, i_, k_] := b[n, i, k] = If[n < 1 || i < 1, 0, If[n == i, 1, Sum[b[n - i, i + j, k], {j, Union[{-k, 0, k}]}]]]; A[n_, k_] := If[n == 0, 1, Sum[b[n, j, k], {j, 1, n}]]; Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 15}] // Flatten (* Jean-François Alcover, Dec 27 2013, translated from Maple *)
CROSSREFS
Column k=0 and main diagonal give: A000005.
Columns k=1, 2 give: A034297, A214253.
Sequence in context: A327742 A103444 A099172 * A214257 A214248 A152719
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Jul 08 2012
STATUS
approved