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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
OFFSET
0,10
LINKS
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
Main diagonal gives A239561.
Sequence in context: A289944 A366747 A055215 * A058398 A091499 A284249
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Mar 21 2014
STATUS
approved