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A243081
Number A(n,k) of compositions of n into parts with multiplicity not larger than k; square array A(n,k), n>=0, k>=0, read by antidiagonals.
21
1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 3, 0, 1, 1, 2, 3, 3, 0, 1, 1, 2, 4, 7, 5, 0, 1, 1, 2, 4, 7, 11, 11, 0, 1, 1, 2, 4, 8, 15, 21, 13, 0, 1, 1, 2, 4, 8, 15, 26, 34, 19, 0, 1, 1, 2, 4, 8, 16, 31, 52, 59, 27, 0, 1, 1, 2, 4, 8, 16, 31, 57, 93, 114, 57, 0, 1, 1, 2, 4, 8, 16, 32, 63, 114, 173, 178, 65, 0
OFFSET
0,13
COMMENTS
A(n,k) is the number of compositions of n avoiding the pattern {1}^(k+1).
LINKS
FORMULA
A(n,k) = Sum_{i=0..k} A242447(n,i).
EXAMPLE
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, ...
0, 1, 1, 1, 1, 1, 1, 1, 1, ...
0, 1, 2, 2, 2, 2, 2, 2, 2, ...
0, 3, 3, 4, 4, 4, 4, 4, 4, ...
0, 3, 7, 7, 8, 8, 8, 8, 8, ...
0, 5, 11, 15, 15, 16, 16, 16, 16, ...
0, 11, 21, 26, 31, 31, 32, 32, 32, ...
0, 13, 34, 52, 57, 63, 63, 64, 64, ...
0, 19, 59, 93, 114, 120, 127, 127, 128, ...
MAPLE
b:= proc(n, i, p, k) option remember; `if`(n=0, p!, `if`(i<1, 0,
add(b(n-i*j, i-1, p+j, k)/j!, j=0..min(n/i, k))))
end:
A:= (n, k)-> `if`(k>=n, `if`(n=0, 1, 2^(n-1)), b(n$2, 0, k)):
seq(seq(A(n, d-n), n=0..d), d=0..14);
MATHEMATICA
b[n_, i_, p_, k_] := b[n, i, p, k] = If[n == 0, p!, If[i<1, 0,
Sum[b[n-i*j, i-1, p+j, k]/j!, {j, 0, Min[n/i, k]}]]];
A[n_, k_] := If[k >= n, If[n == 0, 1, 2^(n-1)], b[n, n, 0, k]];
Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Feb 02 2015, after Alois P. Heinz *)
CROSSREFS
Main diagonal gives A011782.
A(2n,n) gives A232605.
Sequence in context: A096799 A384454 A370292 * A287847 A377666 A336201
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, May 29 2014
STATUS
approved