A220504 Triangle read by rows: T(n,k) is the total number of appearances of k as the smallest part in all partitions of n.

1, 2, 1, 4, 0, 1, 7, 2, 0, 1, 12, 1, 0, 0, 1, 19, 4, 2, 0, 0, 1, 30, 3, 1, 0, 0, 0, 1, 45, 8, 1, 2, 0, 0, 0, 1, 67, 7, 4, 1, 0, 0, 0, 0, 1, 97, 15, 3, 1, 2, 0, 0, 0, 0, 1, 139, 15, 4, 1, 1, 0, 0, 0, 0, 0, 1, 195, 27, 8, 4, 1, 2, 0, 0, 0, 0, 0, 1, 272, 29, 8, 3, 1, 1, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Author

Omar E. Pol, Jan 19 2013

Keywords

• nonn
• tabl

Comments

In other words, T(n,k) is the total number of appearances of k in all partitions of n whose smallest part is k.

The sum of row n equals spt(n), the smallest part partition function (see A092269).

T(n,k) is also the sum of row k in the slice n of tetrahedron A209314.

Examples

			Triangle begins:
    1;
    2,  1;
    4,  0, 1;
    7,  2, 0, 1;
   12,  1, 0, 0, 1;
   19,  4, 2, 0, 0, 1;
   30,  3, 1, 0, 0, 0, 1;
   45,  8, 1, 2, 0, 0, 0, 1;
   67,  7, 4, 1, 0, 0, 0, 0, 1;
   97, 15, 3, 1, 2, 0, 0, 0, 0, 1;
  139, 15, 4, 1, 1, 0, 0, 0, 0, 0, 1;
  195, 27, 8, 4, 1, 2, 0, 0, 0, 0, 0, 1;
  272, 29, 8, 3, 1, 1, 0, 0, 0, 0, 0, 0, 1;
  ...
The partitions of 6 with the smallest part in brackets are
..........................
.                      [6]
..........................
.                  [3]+[3]
..........................
.                   4 +[2]
.              [2]+[2]+[2]
..........................
.                   5 +[1]
.               3 + 2 +[1]
.               4 +[1]+[1]
.           2 + 2 +[1]+[1]
.           3 +[1]+[1]+[1]
.       2 +[1]+[1]+[1]+[1]
.  [1]+[1]+[1]+[1]+[1]+[1]
..........................
There are 19 smallest parts of size 1. Also there are four smallest parts of size 2. Also there are two smallest parts of size 3. There are no smallest part of size 4 or 5. Finally there is only one smallest part of size 6. So row 6 gives 19, 4, 2, 0, 0, 1. The sum of row 6 is 19+4+2+0+0+1 = A092269(6) = 26.
		

Links

• Alois P. Heinz, Rows n = 1..141, flattened

Crossrefs

Columns 1-3: A000070, A087787, A174455.

Row sums give A092269.

Cf. A026794, A182703, A209314.

Programs

Maple

b:= proc(n, i) option remember; local j, r; if n=0 or i<1 then 0
      else `if`(irem(n, i, 'r')=0, [0$(i-1), r], []); for j from 0
      to n/i do zip((x, y)->x+y, %, [b(n-i*j, i-1)], 0) od; %[] fi
    end:
T:= n-> b(n, n):
seq(T(n), n=1..20);  # Alois P. Heinz, Jan 20 2013

Mathematica

b[n_, i_] := b[n, i] = Module[{j, q, r, pc}, If [n == 0 || i<1, 0, {q, r} = QuotientRemainder[n, i]; pc = If[r == 0, Append[Array[0&, i-1], q], {}]; For[j = 0, j <= n/i, j++, pc = Plus @@ PadRight[{pc, b[n-i*j, i-1]}]]; pc]]; T[n_] := b[n, n]; Table[T[n], {n, 1, 20}] // Flatten (* Jean-François Alcover, Jan 30 2014, after Alois P. Heinz *)