A238406 Number T(n,k) of partitions of n into k parts such that every i-th smallest part (counted with multiplicity) is different from i; triangle T(n,k), n>=0, 0<=k<=floor((sqrt(9+8*n)-3)/2) read by rows.

1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 2, 0, 1, 2, 0, 1, 3, 0, 1, 3, 1, 0, 1, 4, 3, 0, 1, 4, 4, 0, 1, 5, 6, 0, 1, 5, 7, 0, 1, 6, 9, 1, 0, 1, 6, 11, 4, 0, 1, 7, 13, 7, 0, 1, 7, 15, 11, 0, 1, 8, 18, 15, 0, 1, 8, 20, 19, 0, 1, 9, 23, 25, 1, 0, 1, 9, 26, 30, 5

Offset: 0

Author

Alois P. Heinz, Feb 26 2014

Keywords

• nonn
• tabf
• look

Examples

			T(10,1) = 1: [10].
T(10,2) = 4: [5,5], [4,6], [3,7], [2,8].
T(10,3) = 3: [3,3,4], [2,4,4], [2,3,5].
Triangle T(n,k) begins:
  1;
  0;
  0, 1;
  0, 1;
  0, 1;
  0, 1, 1;
  0, 1, 2;
  0, 1, 2;
  0, 1, 3;
  0, 1, 3, 1;
  0, 1, 4, 3;
  0, 1, 4, 4;
  0, 1, 5, 6;
  0, 1, 5, 7;
  0, 1, 6, 9, 1;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..500, flattened

Crossrefs

Columns k=0-10 give: A000007, A000012 (for n>1), A004526(n-2) (for n>4), A244239, A244240, A244241, A244242, A244243, A244244, A244245, A244246.

Row sums give A238394.

Cf. A052146.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, (p-> expand(
       x*(p-coeff(p, x, i-1)*x^(i-1))))(b(n-i, i)))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..max(0, degree(p))))(b(n$2)):
seq(T(n), n=0..30);

Mathematica

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, Function[p, Expand[x*(p - Coefficient[p, x, i-1]*x^(i-1))]][b[n-i, i]]]] ]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Max[0, Exponent[p, x]]}]][b[n, n]]; Table[T[n], {n, 0, 30}] // Flatten (* Jean-François Alcover, Feb 08 2017, translated from Maple *)