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A239550 Number A(n,k) of compositions of n such that the first part is 1 and the second differences of the parts are in {-k,...,k}; square array A(n,k), n>=0, k>=0, read by antidiagonals. 13
1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 2, 3, 2, 1, 1, 1, 2, 4, 4, 3, 1, 1, 1, 2, 4, 7, 6, 2, 1, 1, 1, 2, 4, 7, 11, 9, 2, 1, 1, 1, 2, 4, 8, 13, 18, 13, 3, 1, 1, 1, 2, 4, 8, 15, 23, 32, 18, 3, 1, 1, 1, 2, 4, 8, 15, 28, 40, 53, 24, 2, 1, 1, 1, 2, 4, 8, 16, 29, 52, 73, 89, 34, 3 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,10

LINKS

Alois P. Heinz, Antidiagonals n = 0..140, flattened

EXAMPLE

A(6,0) = 3: [1,1,1,1,1,1], [1,2,3], [1,5].

A(5,1) = 4: [1,1,1,1,1], [1,1,1,2], [1,2,2], [1,4].

A(4,2) = 4: [1,1,1,1], [1,1,2], [1,2,1], [1,3].

Square array A(n,k) begins:

1, 1, 1, 1, 1, 1, 1, 1, 1, ...

1, 1, 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, ...

2, 3, 4, 4, 4, 4, 4, 4, 4, ...

2, 4, 7, 7, 8, 8, 8, 8, 8, ...

3, 6, 11, 13, 15, 15, 16, 16, 16, ...

2, 9, 18, 23, 28, 29, 31, 31, 32, ...

2, 13, 32, 40, 52, 56, 60, 61, 63, ...

MAPLE

b:= proc(n, i, j, k) option remember; `if`(n=0, 1,

`if`(i=0, add(b(n-h, j, h, k), h=1..n), add(

b(n-h, j, h, k), h=max(1, 2*j-i-k)..min(n, 2*j-i+k))))

end:

A:= (n, k)-> `if`(n=0, 1, b(n-1, 0, 1, k)):

seq(seq(A(n, d-n), n=0..d), d=0..12);

MATHEMATICA

b[n_, i_, j_, k_] := b[n, i, j, k] = If[n == 0, 1, If[i == 0, Sum[b[n-h, j, h, k], {h, 1, n}], Sum[b[n-h, j, h, k], {h, Max[1, 2*j - i - k], Min[n, 2*j - i + k]}]]] ; A[n_, k_] := If[n == 0, 1, b[n-1, 0, 1, k]]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Jan 22 2015, after Alois P. Heinz *)

CROSSREFS

Columns k=0-10 gives: A129654, A239551, A239552, A239553, A239554, A239555, A239556, A239557, A239558, A239559, A239560.

Main diagonal gives A239561.

Sequence in context: A344318 A289944 A055215 * A058398 A091499 A284249

Adjacent sequences: A239547 A239548 A239549 * A239551 A239552 A239553

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Mar 21 2014

STATUS

approved

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Last modified January 26 22:13 EST 2023. Contains 359836 sequences. (Running on oeis4.)