A216652 Triangular array read by rows: T(n,k) is the number of compositions of n into exactly k distinct parts.

1, 1, 1, 2, 1, 2, 1, 4, 1, 4, 6, 1, 6, 6, 1, 6, 12, 1, 8, 18, 1, 8, 24, 24, 1, 10, 30, 24, 1, 10, 42, 48, 1, 12, 48, 72, 1, 12, 60, 120, 1, 14, 72, 144, 120, 1, 14, 84, 216, 120, 1, 16, 96, 264, 240, 1, 16, 114, 360, 360, 1, 18, 126, 432, 600, 1, 18, 144, 552, 840

Offset: 1

Author

Geoffrey Critzer, Sep 12 2012

Keywords

• nonn
• tabf

Comments

Same as A072574, with zeros dropped. [Joerg Arndt, Oct 20 2012]

Row sums = A032020.

Row n contains A003056(n) = floor((sqrt(8*n+1)-1)/2) terms (number of terms increases by one at each triangular number).

Examples

			Triangle starts:
[ 1]  1;
[ 2]  1;
[ 3]  1, 2;
[ 4]  1, 2;
[ 5]  1, 4;
[ 6]  1, 4, 6;
[ 7]  1, 6, 6;
[ 8]  1, 6, 12;
[ 9]  1, 8, 18;
[10]  1, 8, 24, 24;
[11]  1, 10, 30, 24;
[12]  1, 10, 42, 48;
[13]  1, 12, 48, 72;
[14]  1, 12, 60, 120;
[15]  1, 14, 72, 144, 120;
[16]  1, 14, 84, 216, 120;
[17]  1, 16, 96, 264, 240;
[18]  1, 16, 114, 360, 360;
[19]  1, 18, 126, 432, 600;
[20]  1, 18, 144, 552, 840;
T(5,2) = 4 because we have: 4+1, 1+4, 3+2, 2+3.
		

Links

• Alois P. Heinz, Rows n = 1..500, flattened
• B. Richmond and A. Knopfmacher, Compositions with distinct parts, Aequationes Mathematicae 49 (1995), pp. 86-97.

Crossrefs

Cf. A003056, A008289, A072574, A097910.

Programs

Maple

b:= proc(n, k) option remember; `if`(n<0, 0, `if`(n=0, 1,
      `if`(k<1, 0, b(n, k-1) +b(n-k, k))))
    end:
T:= (n, k)-> b(n-k*(k+1)/2, k)*k!:
seq(seq(T(n, k), k=1..floor((sqrt(8*n+1)-1)/2)), n=1..24);  # Alois P. Heinz, Sep 12 2012

Mathematica

nn=20;f[list_]:=Select[list,#>0&];Map[f,Drop[CoefficientList[Series[ Sum[Product[j y x^j/(1-x^j),{j,1,k}],{k,0,nn}],{x,0,nn}],{x,y}],1]]//Flatten

Formula

G.f.: Sum_{i>=0} Product_{j=1..i} y*j*x^j/(1-x^j).

T(n,k) = A008289(n,k)*k!.