A072575 Triangle T(n,k) of number of compositions (ordered partitions) of n into distinct parts where largest part is exactly k, 1<=k<=n.

1, 0, 1, 0, 2, 1, 0, 0, 2, 1, 0, 0, 2, 2, 1, 0, 0, 6, 2, 2, 1, 0, 0, 0, 8, 2, 2, 1, 0, 0, 0, 6, 8, 2, 2, 1, 0, 0, 0, 6, 8, 8, 2, 2, 1, 0, 0, 0, 24, 12, 8, 8, 2, 2, 1, 0, 0, 0, 0, 30, 14, 8, 8, 2, 2, 1, 0, 0, 0, 0, 30, 36, 14, 8, 8, 2, 2, 1, 0, 0, 0, 0, 24, 36, 38, 14, 8, 8, 2, 2, 1, 0, 0, 0, 0, 24, 54, 42, 38, 14, 8, 8, 2, 2, 1

Offset: 1

Author

Henry Bottomley, Jun 21 2002

Keywords

• nonn
• tabl

Examples

			Rows start:
  1;
  0, 1;
  0, 2, 1;
  0, 0, 2, 1;
  0, 0, 2, 2, 1;
  0, 0, 6, 2, 2, 1;
  0, 0, 0, 8, 2, 2, 1;
  0, 0, 0, 6, 8, 2, 2, 1;
  ...
T(7,4)=8 since 7 can be written as 4+3 =4+2+1 =4+1+2 =3+4 =2+4+1 =2+1+4 =1+4+2 =1+2+4.
		

Links

• Alois P. Heinz, Rows n = 1..141, flattened
• Index entries for sequences related to compositions

Crossrefs

Cf. A026836, A072574. Row sums are A032020. Column sums appear to be A001339 (offset). Starting terms of columns tend towards A072576 as k increases.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i<1, [][], zip((x, y)->x+y, [b(n, i-1)],
      `if`(i>n, [], [0, b(n-i, i-1)]), 0)[]))
    end:
T:= proc(n, k) local l; l:= [b(n-k, k-1)];
       add(l[i]*(i)!, i=1..nops(l))
    end:
seq(seq(T(n, k), k=1..n), n=1..20);  # Alois P. Heinz, Nov 20 2012

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, {1}, If[i<1, {}, Plus @@ PadRight[{b[n, i-1], If[i>n, {}, Join[{0}, b[n-i, i-1]]]}]]]; T[n_, k_] := Module[{l}, l = b[n-k, k-1]; Sum[l[[i]]*i!, {i, 1, Length[l]}]]; Table[Table [T[n, k], {k, 1, n}], {n, 1, 20}] // Flatten (* Jean-François Alcover, Jan 31 2014, after Alois P. Heinz *)