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

1,2

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.

Links

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

Example

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.

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 *)

Crossrefs

Columns 1-3: A000070, A087787, A174455.

Row sums give A092269.

Cf. A026794, A182703, A209314.

Sequence in context: A091604 A200192 A137629 * A087569 A341509 A048614

Adjacent sequences: A220501 A220502 A220503 * A220505 A220506 A220507

Keyword

nonn,tabl

Author

Omar E. Pol, Jan 19 2013

Status

approved