A240315 Triangular array read by rows: T(n,k) is the number of compositions of n into exactly k parts in which no part is unique (each part occurs at least twice).

1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 6, 0, 1, 0, 0, 0, 0, 0, 10, 0, 1, 0, 0, 1, 0, 7, 10, 15, 0, 1, 0, 0, 0, 1, 0, 10, 20, 21, 0, 1, 0, 0, 1, 0, 12, 1, 30, 35, 28, 0, 1, 0, 0, 0, 0, 0, 20, 0, 56, 56, 36, 0, 1, 0, 0, 1, 1, 13, 10, 126, 21, 98, 84, 45, 0, 1

Offset: 0

Author

Geoffrey Critzer, Apr 03 2014

Keywords

• nonn
• tabl

Comments

Row sums = A240085.

Examples

			Triangle begins:
  1;
  0, 0;
  0, 0, 1;
  0, 0, 0, 1;
  0, 0, 1, 0,  1;
  0, 0, 0, 0,  0,  1;
  0, 0, 1, 1,  6,  0,   1;
  0, 0, 0, 0,  0, 10,   0,  1;
  0, 0, 1, 0,  7, 10,  15,  0,  1;
  0, 0, 0, 1,  0, 10,  20, 21,  0,  1;
  0, 0, 1, 0, 12,  1,  30, 35, 28,  0,  1;
  0, 0, 0, 0,  0, 20,   0, 56, 56, 36,  0, 1;
  0, 0, 1, 1, 13, 10, 126, 21, 98, 84, 45, 0, 1;
  ...
T(8,4) = 7 because we have: 3+3+1+1, 3+1+3+1, 3+1+1+3, 1+3+3+1, 1+3+1+3, 1+1+3+3, 2+2+2+2.
		

References

• S. Heubach and T. Mansour, Combinatorics of Compositions and Words, Chapman and Hall, 2009 page 87.

Links

• Alois P. Heinz, Rows n = 0..140, flattened

Programs

Maple

b:= proc(n, i, t) option remember; `if`(n=0, t!, `if`(i<1, 0,
      expand(b(n, i-1, t)+add(x^j*b(n-i*j, i-1, t+j)/j!, j=2..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2, 0)):
seq(T(n), n=0..14);  # Alois P. Heinz, Apr 03 2014

Mathematica

nn=10;Table[Take[Transpose[Range[0,nn]!CoefficientList[Series[ Product[Exp[x^i y]-x^i y,{i,1,nn}],{y,0,nn}],{y,x}]],nn+1][[j,Range[1,j]]],{j,1,nn}]//Grid

Formula

Product_{i>=1} exp(x^i*y) - x^i*y = Sum_{k>=0} A_k(x)*y^k/k!, where A_k(x) is the o.g.f. for the number of compositions of n into k parts in which no part is unique. In other words, A_k(x) is the o.g.f. for column k.