A194549 Triangle read by rows: T(n,k) = Dyson's rank of the k-th partition of n that does not contain 1 as a part, with partitions in lexicographic order.

1, 1, 2, 0, 3, 1, 4, -1, 2, 1, 5, 0, 3, 2, 6, -2, 1, 0, 4, 3, 2, 7, -1, 2, 1, 5, 0, 4, 3, 8, -3, 0, -1, 3, 2, 1, 6, 1, 5, 4, 3, 9, -2, 1, 0, 4, -1, 3, 2, 7, 2, 1, 6, 5, 4, 10, -4, -1, -2, 2, 1, 0, 5, 0, 4, 3, 2, 8, -1, 3, 2, 7, 1, 6, 5, 4, 11, -3, 0, -1, 3, -2

Offset: 1

Author

Omar E. Pol, Dec 11 2011

Keywords

• sign
• tabf
• look

Examples

			Written as a triangle:
  1;
  1;
  2;
  0,3;
  1,4;
  -1,2,1,5;
  0,3,2,6;
  -2,1,0,4,3,2,7;
  -1,2,1,5,0,4,3,8;
  -3,0,-1,3,2,1,6,1,5,4,3,9;
  -2,1,0,4,-1,3,2,7,2,1,6,5,4,10;
  -4,-1,-2,2,1,0,5,0,4,3,2,8,-1,3,2,7,1,6,5,4,11;
		

Links

• Alois P. Heinz, Rows n = 1..33, flattened

Crossrefs

The sum of row n is A000041(n-1). Row n has length A187219(n).

Cf. A002865, A135010, A138121, A194546, A194547, A194548.

Programs

Maple

T:= proc(n) local b, l;
      b:= proc(n, i, t)
            if n=0 then l:=l, i-t
          elif i>n then
          else b(n-i, i, t+1); b(n, i+1, t)
            fi
          end;
      if n<2 then 1 else l:= NULL; b(n, 2, 0); l fi
    end:
seq(T(n), n=1..13); # Alois P. Heinz, Dec 20 2011

Mathematica

T[n_] := Module[{b, l}, b[n0_, i_, t_] :=
     If[n0 == 0, l = Append[l, i-t],
     If[i>n0, , b[n0-i, i, t+1]; b[n0, i+1, t]]];
     If[n<2, {1}, l = {}; b[n, 2, 0]; l]];
Table[T[n], {n, 1, 13}]  // Flatten (* Jean-François Alcover, Mar 05 2021, after Alois P. Heinz *)

Formula

a(n) = A141285(n) - A194548(n).

Extensions

More terms from Alois P. Heinz, Dec 20 2011