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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.

%I A243081 #26 Nov 03 2021 08:34:42
%S A243081 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 A243081 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 A243081 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 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.
%C A243081 A(n,k) is the number of compositions of n avoiding the pattern {1}^(k+1).
%H A243081 Alois P. Heinz, <a href="/A243081/b243081.txt">Rows n = 0..140, flattened</a>
%F A243081 A(n,k) = Sum_{i=0..k} A242447(n,i).
%e A243081 Square array A(n,k) begins:
%e A243081   1,  1,  1,  1,   1,   1,   1,   1,   1, ...
%e A243081   0,  1,  1,  1,   1,   1,   1,   1,   1, ...
%e A243081   0,  1,  2,  2,   2,   2,   2,   2,   2, ...
%e A243081   0,  3,  3,  4,   4,   4,   4,   4,   4, ...
%e A243081   0,  3,  7,  7,   8,   8,   8,   8,   8, ...
%e A243081   0,  5, 11, 15,  15,  16,  16,  16,  16, ...
%e A243081   0, 11, 21, 26,  31,  31,  32,  32,  32, ...
%e A243081   0, 13, 34, 52,  57,  63,  63,  64,  64, ...
%e A243081   0, 19, 59, 93, 114, 120, 127, 127, 128, ...
%p A243081 b:= proc(n, i, p, k) option remember; `if`(n=0, p!, `if`(i<1, 0,
%p A243081       add(b(n-i*j, i-1, p+j, k)/j!, j=0..min(n/i, k))))
%p A243081     end:
%p A243081 A:= (n, k)-> `if`(k>=n, `if`(n=0, 1, 2^(n-1)), b(n$2, 0, k)):
%p A243081 seq(seq(A(n, d-n), n=0..d), d=0..14);
%t A243081 b[n_, i_, p_, k_] := b[n, i, p, k] = If[n == 0, p!, If[i<1, 0,
%t A243081      Sum[b[n-i*j, i-1, p+j, k]/j!, {j, 0, Min[n/i, k]}]]];
%t A243081 A[n_, k_] := If[k >= n, If[n == 0, 1, 2^(n-1)], b[n, n, 0, k]];
%t A243081 Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* _Jean-François Alcover_, Feb 02 2015, after _Alois P. Heinz_ *)
%Y A243081 Columns k=0-10 give: A000007, A032020, A232432, A232464, A243082, A243083, A243084, A243085, A243086, A243087, A243088.
%Y A243081 Main diagonal gives A011782.
%Y A243081 A(2n,n) gives A232605.
%K A243081 nonn,tabl
%O A243081 0,13
%A A243081 _Alois P. Heinz_, May 29 2014