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

0,8

Comments

Row sums = A000041, the partition numbers.

Links

Table of n, a(n) for n=0..76.

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.

Example

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.

Sequence in context: A091423 A221914 A264401 * A029251 A263073 A133091

Adjacent sequences: A173301 A173302 A173303 * A173305 A173306 A173307

Keyword

nonn,tabf,uned

Author

Gary W. Adamson, Feb 15 2010

Status

approved