A173304 Triangle generated from the array in A173302 (partition numbers starting new rows at n = 1, 3, 7, 15, ...).

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 2, 3, 2, 4, 4, 3, 4, 6, 4, 7, 8, 6, 1, 8, 11, 9, 2, 12, 15, 12, 3, 14, 20, 17, 5, 21, 26, 23, 7, 24, 35, 31, 11, 34, 45, 41, 15, 41, 58, 55, 21, 1, 55, 75, 71, 29, 1, 66, 96, 93, 40, 2, 88, 121, 120, 53, 3, 105, 154, 154, 72, 5, 137, 193, 196, 94, 7

Offset: 0

Author

Gary W. Adamson, Feb 15 2010

Keywords

• nonn
• tabf
• uned

Comments

Row sums = A000041, the partition numbers.

Examples

			The finite difference array starts:
  1, 1, 1, 1, 2, 2, 4, 4, 7,  8, 12, 14, 21, 24, ...; = A002865 (a variant)
        1, 1, 2, 3, 4, 6, 8, 11, 15, 20, 26, 35, ...; = A027336
           1, 1, 2, 3, 4, 6,  9, 12, 17, 23, 31, ...; = A017338
                       1, 1,  2,  3,  5,  7, 11, ...; = A027342
  ...
Last, columns of the array become rows of triangle A173304:
    1;
    1;
    1,   1;
    2,   2,   1;
    2,   3,   2;
    4,   4,   3;
    4,   6,   4,  1;
    7,   8,   6,  1;
    8,  11,   9,  2;
   12,  15,  12,  3;
   14,  20,  17,  5;
   21,  26,  23,  7;
   24,  35,  31, 11;
   34,  45,  41, 15;
   41,  58,  55, 21, 1;
   55,  75,  71, 29, 1;
   66,  96,  93, 40, 2;
   88, 121, 120, 53, 3;
  105, 154, 154, 72, 5;
  137, 193, 196, 94, 7;
  ...
		

Crossrefs

Cf. A000041, A173301, A173302, A173303.

Formula

The generating array is in A173302.

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, ...

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, ...

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, ...

1, 1, 2, 3, 5, 7, 11, 15, 22, ...

1, ...

...

Take finite differences from the bottom, creating a new array in which rows are A002865 (a slight variant), A027336, A027338, A027342, ...; i.e., the numbers of partitions of n that do not contain (1, 2, 4, 8, ...) as a part.