A299778 Irregular triangle read by rows: T(n,k) is the part that is adjacent to the k-th peak of the largest Dyck path of the symmetric representation of sigma(n), or T(n,k) = 0 if the mentioned part is already associated to a previous peak or if there is no part adjacent to the k-th peak, with n >= 1, k >= 1.

1, 3, 2, 2, 7, 0, 3, 3, 12, 0, 0, 4, 0, 4, 15, 0, 0, 5, 3, 5, 9, 0, 9, 0, 6, 0, 0, 6, 28, 0, 0, 0, 7, 0, 0, 7, 12, 0, 12, 0, 8, 8, 0, 0, 8, 31, 0, 0, 0, 0, 9, 0, 0, 0, 9, 39, 0, 0, 0, 0, 10, 0, 0, 0, 10, 42, 0, 0, 0, 0, 11, 5, 0, 5, 0, 11, 18, 0, 0, 0, 18, 0, 12, 0, 0, 0, 0, 12, 60, 0, 0, 0, 0, 0, 13, 0, 5, 0, 0, 13

Offset

1,2

Comments

For the definition of "part" of the symmetric representation of sigma see A237270.

For more information about the mentioned Dyck paths see A237593.

Links

Table of n, a(n) for n=1..100.

Index entries for sequences related to sigma(n)

Example

Triangle begins (rows 1..28):
   1;
   3;
   2,  2;
   7,  0;
   3,  3;
  12,  0,  0;
   4,  0,  4;
  15,  0,  0;
   5,  3,  5;
   9,  0,  9,  0;
   6,  0,  0,  6;
  28,  0,  0,  0;
   7,  0,  0,  7;
  12,  0, 12,  0;
   8,  8,  0,  0,  8;
  31,  0,  0,  0,  0;
   9,  0,  0,  0,  9;
  39,  0,  0,  0,  0;
  10,  0,  0,  0, 10;
  42,  0,  0,  0,  0;
  11,  5,  0,  5,  0, 11;
  18,  0,  0,  0, 18,  0;
  12,  0,  0,  0,  0, 12;
  60,  0,  0,  0,  0,  0;
  13,  0,  5,  0,  0, 13;
  21,  0,  0,  0  21,  0;
  14,  6,  0,  6,  0, 14;
  56,  0,  0,  0,  0,  0,  0;
  ...
Illustration of first 50 terms (rows 1..16 of triangle) in an irregular spiral which can be find in the top view of the pyramid described in A244050:
.
.               12 _ _ _ _ _ _ _ _
.                 |  _ _ _ _ _ _ _|_ _ _ _ _ _ _ 7
.                 | |             |_ _ _ _ _ _ _|
.              0 _| |                           |
.               |_ _|9 _ _ _ _ _ _              |_ _ 0
.         12 _ _|     |  _ _ _ _ _|_ _ _ _ _ 5      |_ 0
.    0 _ _ _| |    0 _| |         |_ _ _ _ _|         |
.     |  _ _ _|  9 _|_ _|                   |_ _ 3    |_ _ _ 7
.     | |    0 _ _| |   12 _ _ _ _          |_  |         | |
.     | |     |  _ _|  0 _|  _ _ _|_ _ _ 3    |_|_ _ 5    | |
.     | |     | |    0 _|   |     |_ _ _|         | |     | |
.     | |     | |     |  _ _|           |_ _ 3    | |     | |
.     | |     | |     | |    3 _ _        | |     | |     | |
.     | |     | |     | |     |  _|_ 1    | |     | |     | |
.    _|_|    _|_|    _|_|    _|_| |_|    _|_|    _|_|    _|_|    _
.   | |     | |     | |     | |         | |     | |     | |     | |
.   | |     | |     | |     |_|_ _     _| |     | |     | |     | |
.   | |     | |     | |    2  |_ _|_ _|  _|     | |     | |     | |
.   | |     | |     |_|_     2    |_ _ _|  0 _ _| |     | |     | |
.   | |     | |    4    |_               7 _|  _ _|0    | |     | |
.   | |     |_|_ _     0  |_ _ _ _        |  _|    _ _ _| |     | |
.   | |    6      |_      |_ _ _ _|_ _ _ _| |  0 _|    _ _|0    | |
.   |_|_ _ _     0  |_   4        |_ _ _ _ _|  _|     |    _ _ _| |
.  8      | |_ _   0  |                     15|      _|   |  _ _ _|
.         |_    |     |_ _ _ _ _ _            |  _ _|  0 _| |      0
.        8  |_  |_    |_ _ _ _ _ _|_ _ _ _ _ _| |    0 _|  _|
.          0  |_ _|  6            |_ _ _ _ _ _ _|  _ _|  _|  0
.            0    |                             28|  _ _|  0
.                 |_ _ _ _ _ _ _ _                | |    0
.                 |_ _ _ _ _ _ _ _|_ _ _ _ _ _ _ _| |
.                8                |_ _ _ _ _ _ _ _ _|
.                                                    31
.
The diagram contains A237590(16) = 27 parts.
For the construction of the spiral see A239660.

Crossrefs

Row sums give A000203.

Row n has length A003056(n).

Column k starts in row A000217(k).

Nonzero terms give A237270.

The number of nonzero terms in row n is A237271(n).

Column 1 is A241838.

The triangle with n rows contain A237590(n) nonzero terms.

Cf. A296508 (analog for subparts).

Cf. A024916, A196020, A235791, A236104, A237048, A237270, A237271, A237591, A237593, A239657, A239660, A239931-A239934, A240542, A244050, A245092, A250068, A250070, A261699, A262626, A279387, A279388, A279391, A280850, A280851.

Sequence in context: A354002 A280850 A296508 * A302248 A235773 A089327

Adjacent sequences: A299775 A299776 A299777 * A299779 A299780 A299781

Keyword

nonn,tabf

Author

Omar E. Pol, Apr 03 2018

Status

approved