A194548 Triangle read by rows: T(n,k) = number of parts in the k-th partition of n that does not contain 1 as a part, with partitions in lexicographic order.

0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 2, 2, 1, 4, 3, 3, 2, 2, 2, 1, 4, 3, 3, 2, 3, 2, 2, 1, 5, 4, 4, 3, 3, 3, 2, 3, 2, 2, 2, 1, 5, 4, 4, 3, 4, 3, 3, 2, 3, 3, 2, 2, 2, 1, 6, 5, 5, 4, 4, 4, 3, 4, 3, 3, 3, 2, 4, 3, 3, 2, 3, 2, 2, 2, 1, 6, 5, 5, 4, 5, 4, 4, 3, 4, 4, 3, 3, 3, 2, 4, 3, 3, 3, 2, 3, 2, 2, 2, 1

Offset: 1

Author

Omar E. Pol, Dec 11 2011

Keywords

• nonn
• tabf

Examples

			Written as a triangle:
  0;
  1;
  1;
  2,1;
  2,1;
  3,2,2,1;
  3,2,2,1;
  4,3,3,2,2,2,1;
  4,3,3,2,3,2,2,1;
  5,4,4,3,3,3,2,3,2,2,2,1;
  5,4,4,3,4,3,3,2,3,3,2,2,2,1;
  6,5,5,4,4,4,3,4,3,3,3,2,4,3,3,2,3,2,2,2,1;
  6,5,5,4,5,4,4,3,4,4,3,3,3,2,4,3,3,3,2,3,2,2,2,1;
		

Links

• Alois P. Heinz, Rows n = 1..33, flattened
• Tilman Piesk, Table for A194602, showing the non-one addends.

Crossrefs

Row sums give A138135. Row n has length A187219(n).

Cf. A002865, A135010, A138121, A193173, A194546, A194547, A194549.

Programs

Maple

T:= proc(n) local b, l;
      b:= proc(n, i, t)
            if n=0 then l:=l, t
          elif i>n then
          else b(n-i, i, t+1); b(n, i+1, t)
            fi
          end;
      if n<2 then 0 else l:= NULL; b(n, 2, 0); l fi
    end:
seq(T(n), n=1..15);  # Alois P. Heinz, Dec 19 2011

Mathematica

T[n_] := Module[{b, l}, b[n0_, i_, t_] :=
     If[n0==0, l = Append[l, t],
     If[i>n0, , b[n0-i, i, t+1]; b[n0, i+1, t]]];
     If[n<2, {0}, l = {}; b[n, 2, 0]; l]];
Table[T[n], {n, 1, 15}]  // Flatten (* Jean-François Alcover, Mar 05 2021, after Alois P. Heinz *)

Extensions

More terms from Alois P. Heinz, Dec 19 2011