A242451 Number T(n,k) of compositions of n in which the minimal multiplicity of parts equals k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
1, 0, 1, 0, 1, 1, 0, 3, 0, 1, 0, 6, 1, 0, 1, 0, 15, 0, 0, 0, 1, 0, 23, 7, 1, 0, 0, 1, 0, 53, 10, 0, 0, 0, 0, 1, 0, 94, 32, 0, 1, 0, 0, 0, 1, 0, 203, 31, 21, 0, 0, 0, 0, 0, 1, 0, 404, 71, 35, 0, 1, 0, 0, 0, 0, 1, 0, 855, 77, 91, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1648, 222, 105, 71, 0, 1, 0, 0, 0, 0, 0, 1
Offset: 0
Examples
T(5,1) = 15: [1,1,1,2], [1,1,2,1], [1,2,1,1], [2,1,1,1], [1,2,2], [2,1,2], [2,2,1], [1,1,3], [1,3,1], [3,1,1], [2,3], [3,2], [1,4], [4,1], [5]. T(6,2) = 7: [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]. T(6,3) = 1: [2,2,2]. Triangle T(n,k) begins: 1; 0, 1; 0, 1, 1; 0, 3, 0, 1; 0, 6, 1, 0, 1; 0, 15, 0, 0, 0, 1; 0, 23, 7, 1, 0, 0, 1; 0, 53, 10, 0, 0, 0, 0, 1; 0, 94, 32, 0, 1, 0, 0, 0, 1; 0, 203, 31, 21, 0, 0, 0, 0, 0, 1; 0, 404, 71, 35, 0, 1, 0, 0, 0, 0, 1;
Links
- Alois P. Heinz, Rows n = 0..140, flattened
Crossrefs
Programs
-
Maple
b:= proc(n, i, p, k) option remember; `if`(n=0, p!, `if`(i<1, 0, b(n, i-1, p, k) +add(b(n-i*j, i-1, p+j, k)/j!, j=max(1, k)..floor(n/i)))) end: T:= (n, k)-> b(n$2, 0, k) -`if`(n=0 and k=0, 0, b(n$2, 0, k+1)): seq(seq(T(n, k), k=0..n), n=0..14); -
Mathematica
b[n_, i_, p_, k_] := b[n, i, p, k] = If[n == 0, p!, If[i < 1, 0, b[n, i - 1, p, k] + Sum[b[n - i*j, i - 1, p + j, k]/j!, {j, Max[1, k], Floor[n/i]}]]]; T[n_, k_] := b[n, n, 0, k] - If[n == 0 && 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 27 2015, after Alois P. Heinz *)
Comments