+----------------------------------------------------+
          |                                                    |
          |  t(n,k) = 1 + Sum(t(i,j): i>j>k & i+j=n), 0<=k<=n  |
          |                                                    |
          +----------------------------------------------------+



+----+                                 k = 0
|  n |                                    /  1
+----+                                      /  2
|  0 |  . . . . . . . . . . . . .      1      /  3
|  1 |  . . . . . . . . . . . .      1   1      /  4
|  2 |  . . . . . . . . . . .      1   1   1      /  5
|  3 |  . . . . . . . . . .      2   1   1   1      /  6
|  4 |  . . . . . . . . .      2   1   1   1   1      /  7
|  5 |  . . . . . . . .      3   2   1   1   1   1      /  8
|  6 |  . . . . . . .      4   2   1   1   1   1   1      /  9
|  7 |  . . . . . .      5   3   2   1   1   1   1   1      / 10
|  8 |  . . . . .      6   3   2   1   1   1   1   1   1      / 11
|  9 |  . . . .      8   5   3   2   1   1   1   1   1   1      / 12
| 10 |  . . .     10   5   3   2   1   1   1   1   1   1   1      / 13
| 11 |  . .     12   7   4   3   2   1   1   1   1   1   1   1      /  14
| 12 |  .     15   8   4   3   2   1   1   1   1   1   1   1   1      /  15
| 13 |      18  10   6   4   3   2   1   1   1   1   1   1   1   1       /
| 14 |    22  12   7   4   3   2   1   1   1   1   1   1   1   1   1     
| 15 |  27  15   9   6   4   3   2   1   1   1   1   1   1   1   1   1


Properties:
[1] t(n,k) > 1 iff 0 <= 2*k+1 < n
[2] t(n,k) <= t(n+1,k)
[3] t(n,k+1) <= t(n,k)


Proposition: t(n,k) = 1 + Sum(t(i,j): i>j>k & i+j=n), 0<=k<=n,  is the number 
of all partitions of n into distinct numbers notless k.

Proof outline: The "1" counts {n}, the trivial unique partition,  whereas the 
"Sum" counts recursevly the proper partitions, having more than one  member 
(see example for illustration).


Corollary: t(n,0) is the number of all partitions of n into distinct  positive 
integers; A000009(n) = t(n,0).




E x a m p l e:   n = 13

t(13,0) = 1 + t(12,1) + t(11,2) + t(10,3) + t(9,4) + t(8,5) + t(7,6);
        t(12,1) = 1 + t(10,2) + t(9,3) + t(8,4) + t(7,5);
                t(10,2) = 1 + t(7,3) + t(6,4);
                        t(7,3) = 1;  t(6,4) = 1;
                t(9,3) = 1 + t(5,4);
                       t(5,4) = 1;
                t(8,4) = 1; t(7,5) = 1
        t(11,2) = 1 + t(8,3) + t(7,4) + t(6,5);
                t(8,3) = 1;  t(7,4) = 1;  t(6,5) = 1
        t(10,3) = 1 + t(6,4)
                t(6,4) = 1
        t(9,4) = 1;  t(8,5) = 1; t(7,6) = 1

t(13,0) = 1 + ((1 + ((1+1+1)+(1+1)+1+1)) + (1+1+1+1) + (1+1) + 1 + 1  + 1) 
        = 1 + 8 + 4 + 2 + 1 + 1 + 1 = 18


+----+                                 k = 0
|  n |                                    /  1
+----+                                      /  2
|  0 |  . . . . . . . . . . . . .      _      /  3
|  1 |  . . . . . . . . . . . .      _   _      /  4
|  2 |  . . . . . . . . . . .      _   _   _      /  5
|  3 |  . . . . . . . . . .      _   _   _   _      /  6
|  4 |  . . . . . . . . .      _   _   _   _   _      /  7
|  5 |  . . . . . . . .      _   _   _   _   1   _      /  8
|  6 |  . . . . . . .      _   _   _   _   1   1   _      /  9
|  7 |  . . . . . .      _   _   _   1   1   1   1   _      / 10
|  8 |  . . . . .      _   _   _   1   1   1   _   _   _      / 11
|  9 |  . . . .      _   _   _   2   1   _   _   _   _   _      / 12
| 10 |  . . .      _   _   3   2   _   _   _   _   _   _   _      / 13
| 11 |  . .      _   _   4   _   _   _   _   _   _   _   _   _      /  14
| 12 |  .      _   8   _   _   _   _   _   _   _   _   _   _   _      /  15
| 13 |      18   _   _   _   _   _   _   _   _   _   _   _   _   _       /
| 14 |     _   _   _   _   _   _   _   _   _   _   _   _   _   _   _



 1)    13 :     [x x x x x x x x x x x x x]
 2)    12+1:    [x x x x x x x x x x x x]  [x]
 3)    10+2+1:  [x x x x x x x x x x]      [x x]       [x]
 4)    7+3+2+1: [x x x x x x x]            [x x x]     [x x]   [x]
 5)    6+4+2+1: [x x x x x x]              [x x x x]   [x x]   [x]
 6)    9+3+1:   [x x x x x x x x x]        [x x x]     [x]
 7)    5+4+3+1: [x x x x x]                [x x x x]   [x x x] [x]
 8)    8+4+1:   [x x x x x x x x]          [x x x x]   [x]
 9)    7+5+1:   [x x x x x x x]            [x x x x x] [x]
10)    11+2:    [x x x x x x x x x x x]    [x x]
11)    8+3+2:   [x x x x x x x x]          [x x x]     [x x]
12)    7+4+2:   [x x x x x x x]            [x x x x]   [x x]
13)    6+5+2:   [x x x x x x]              [x x x x x] [x x]
14)    10+3:    [x x x x x x x x x x]      [x x x]
15)    6+4+3:   [x x x x x x]              [x x x x]   [x x x]
16)    9+4:     [x x x x x x x x x]        [x x x x]
17)    8+5:     [x x x x x x x x]          [x x x x x]
18)    7+6:     [x x x x x x x]            [x x x x x x]