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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.
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%I #26 Nov 03 2021 08:34:42

%S 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,

%T 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,

%U 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

%N 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.

%C A(n,k) is the number of compositions of n avoiding the pattern {1}^(k+1).

%H Alois P. Heinz, <a href="/A243081/b243081.txt">Rows n = 0..140, flattened</a>

%F A(n,k) = Sum_{i=0..k} A242447(n,i).

%e Square array A(n,k) begins:

%e 1, 1, 1, 1, 1, 1, 1, 1, 1, ...

%e 0, 1, 1, 1, 1, 1, 1, 1, 1, ...

%e 0, 1, 2, 2, 2, 2, 2, 2, 2, ...

%e 0, 3, 3, 4, 4, 4, 4, 4, 4, ...

%e 0, 3, 7, 7, 8, 8, 8, 8, 8, ...

%e 0, 5, 11, 15, 15, 16, 16, 16, 16, ...

%e 0, 11, 21, 26, 31, 31, 32, 32, 32, ...

%e 0, 13, 34, 52, 57, 63, 63, 64, 64, ...

%e 0, 19, 59, 93, 114, 120, 127, 127, 128, ...

%p b:= proc(n, i, p, k) option remember; `if`(n=0, p!, `if`(i<1, 0,

%p add(b(n-i*j, i-1, p+j, k)/j!, j=0..min(n/i, k))))

%p end:

%p A:= (n, k)-> `if`(k>=n, `if`(n=0, 1, 2^(n-1)), b(n$2, 0, k)):

%p seq(seq(A(n, d-n), n=0..d), d=0..14);

%t b[n_, i_, p_, k_] := b[n, i, p, k] = If[n == 0, p!, If[i<1, 0,

%t Sum[b[n-i*j, i-1, p+j, k]/j!, {j, 0, Min[n/i, k]}]]];

%t A[n_, k_] := If[k >= n, If[n == 0, 1, 2^(n-1)], b[n, n, 0, k]];

%t Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* _Jean-François Alcover_, Feb 02 2015, after _Alois P. Heinz_ *)

%Y Columns k=0-10 give: A000007, A032020, A232432, A232464, A243082, A243083, A243084, A243085, A243086, A243087, A243088.

%Y Main diagonal gives A011782.

%Y A(2n,n) gives A232605.

%K nonn,tabl

%O 0,13

%A _Alois P. Heinz_, May 29 2014