%I #27 Oct 25 2018 14:45:59
%S 1,0,1,0,1,1,0,3,0,1,0,3,4,0,1,0,5,6,4,0,1,0,11,10,5,5,0,1,0,13,21,18,
%T 5,6,0,1,0,19,40,34,21,6,7,0,1,0,27,87,59,40,27,7,8,0,1,0,57,121,132,
%U 100,49,35,8,9,0,1,0,65,219,272,210,131,63,44,9,10,0,1
%N Number T(n,k) of compositions of n in which the maximal multiplicity of parts equals k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%C T(0,0) = 1 by convention. T(n,k) counts the compositions of n in which at least one part has multiplicity k and no part has a multiplicity larger than k.
%H Alois P. Heinz, <a href="/A242447/b242447.txt">Rows n = 0..140, flattened</a>
%e T(6,1) = 11: [1,2,3], [1,3,2], [2,1,3], [2,3,1], [3,1,2], [3,2,1], [2,4], [4,2], [1,5], [5,1], [6].
%e T(6,2) = 10: [1,1,2,2], [1,2,1,2], [1,2,2,1], [2,1,1,2], [2,1,2,1], [2,2,1,1], [3,3], [1,1,4], [1,4,1], [4,1,1].
%e T(6,3) = 5: [2,2,2], [1,1,1,3], [1,1,3,1], [1,3,1,1], [3,1,1,1].
%e T(6,4) = 5: [1,1,1,1,2], [1,1,1,2,1], [1,1,2,1,1], [1,2,1,1,1], [2,1,1,1,1].
%e T(6,6) = 1: [1,1,1,1,1,1].
%e Triangle T(n,k) begins:
%e 1;
%e 0, 1;
%e 0, 1, 1;
%e 0, 3, 0, 1;
%e 0, 3, 4, 0, 1;
%e 0, 5, 6, 4, 0, 1;
%e 0, 11, 10, 5, 5, 0, 1;
%e 0, 13, 21, 18, 5, 6, 0, 1;
%e 0, 19, 40, 34, 21, 6, 7, 0, 1;
%e 0, 27, 87, 59, 40, 27, 7, 8, 0, 1;
%e 0, 57, 121, 132, 100, 49, 35, 8, 9, 0, 1;
%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 T:= (n, k)-> b(n$2, 0, k) -`if`(k=0, 0, b(n$2, 0, k-1)):
%p seq(seq(T(n, k), k=0..n), n=0..14);
%t 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]}]]]; T[n_, k_] := b[n, n, 0, k] - If[k == 0, 0, b[n, n, 0, k-1]]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 14}] // Flatten (* _Jean-François Alcover_, Jan 22 2015, after _Alois P. Heinz_ *)
%Y Columns k=0-10 give: A000007, A032020 (for n>0), A243119, A243120, A243121, A243122, A243123, A243124, A243125, A243126, A243127.
%Y T(2n,n) = A232665(n).
%Y Row sums give A011782.
%Y Cf. A242451 (the same for minimal multiplicity).
%K nonn,tabl
%O 0,8
%A _Alois P. Heinz_, May 15 2014