A173303 -id:A173303 - cp's OEIS frontend

A173302 Irregular triangle by rows, generated from an array with A000041 starting again at columns (2^n - 1), n=1,2,3,...

1, 1, 1, 2, 1, 3, 2, 1, 5, 3, 1, 7, 5, 2, 11, 7, 3, 15, 11, 5, 1, 22, 15, 7, 1, 30, 22, 11, 2, 42, 30, 15, 3, 56, 42, 22, 5, 77, 56, 30, 7, 101, 77, 42, 11, 135, 101, 56, 15, 176, 135, 77, 22, 1, 231, 176, 101, 30, 1, 297, 231, 135, 42, 2

Offset: 0

Gary W. Adamson, Feb 15 2010

Sum of finite difference terms of the array create triangle A173304.

Row sums = A173303.

			The generating array =
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,...
...then columns become the rows of triangle A173302:
1;
1, 1;
2, 1;
3, 2, 1;
5, 3, 1;
7, 5, 2;
11, 7, 3;
15, 11, 5, 1;
22, 15, 7, 1;
30, 22, 11, 2;
42, 30, 15, 3;
56, 42, 22, 5;
77, 56, 30, 7;
101, 77, 4, 11;
135, 101, 56, 15;
176, 135, 77, 22, 1;
...

Cf. A000041, A173301, A173303, A173304.

Create an array with A000041 in row 0, then A000041 starts anew under successive columns (1, 3, 7, 15, 31,...). (Cf. A173301).

Columns of the array = rows of A173302.

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

Original entry on oeis.org

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

Gary W. Adamson, Feb 15 2010

Row sums = A000041, the partition numbers.

			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;
  ...

Cf. A000041, A173301, A173302, A173303.

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.

cp's OEIS Frontend

A173302 Irregular triangle by rows, generated from an array with A000041 starting again at columns (2^n - 1), n=1,2,3,...

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