A299778 - cp's OEIS frontend

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.

%I A299778 #23 Jun 19 2019 17:57:24
%S A299778 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,
%T A299778 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,
%U A299778 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
%N 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.
%C A299778 For the definition of "part" of the symmetric representation of sigma see A237270.
%C A299778 For more information about the mentioned Dyck paths see A237593.
%H A299778 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%e A299778 Triangle begins (rows 1..28):
%e A299778    1;
%e A299778    3;
%e A299778    2,  2;
%e A299778    7,  0;
%e A299778    3,  3;
%e A299778   12,  0,  0;
%e A299778    4,  0,  4;
%e A299778   15,  0,  0;
%e A299778    5,  3,  5;
%e A299778    9,  0,  9,  0;
%e A299778    6,  0,  0,  6;
%e A299778   28,  0,  0,  0;
%e A299778    7,  0,  0,  7;
%e A299778   12,  0, 12,  0;
%e A299778    8,  8,  0,  0,  8;
%e A299778   31,  0,  0,  0,  0;
%e A299778    9,  0,  0,  0,  9;
%e A299778   39,  0,  0,  0,  0;
%e A299778   10,  0,  0,  0, 10;
%e A299778   42,  0,  0,  0,  0;
%e A299778   11,  5,  0,  5,  0, 11;
%e A299778   18,  0,  0,  0, 18,  0;
%e A299778   12,  0,  0,  0,  0, 12;
%e A299778   60,  0,  0,  0,  0,  0;
%e A299778   13,  0,  5,  0,  0, 13;
%e A299778   21,  0,  0,  0  21,  0;
%e A299778   14,  6,  0,  6,  0, 14;
%e A299778   56,  0,  0,  0,  0,  0,  0;
%e A299778   ...
%e A299778 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:
%e A299778 .
%e A299778 .               12 _ _ _ _ _ _ _ _
%e A299778 .                 |  _ _ _ _ _ _ _|_ _ _ _ _ _ _ 7
%e A299778 .                 | |             |_ _ _ _ _ _ _|
%e A299778 .              0 _| |                           |
%e A299778 .               |_ _|9 _ _ _ _ _ _              |_ _ 0
%e A299778 .         12 _ _|     |  _ _ _ _ _|_ _ _ _ _ 5      |_ 0
%e A299778 .    0 _ _ _| |    0 _| |         |_ _ _ _ _|         |
%e A299778 .     |  _ _ _|  9 _|_ _|                   |_ _ 3    |_ _ _ 7
%e A299778 .     | |    0 _ _| |   12 _ _ _ _          |_  |         | |
%e A299778 .     | |     |  _ _|  0 _|  _ _ _|_ _ _ 3    |_|_ _ 5    | |
%e A299778 .     | |     | |    0 _|   |     |_ _ _|         | |     | |
%e A299778 .     | |     | |     |  _ _|           |_ _ 3    | |     | |
%e A299778 .     | |     | |     | |    3 _ _        | |     | |     | |
%e A299778 .     | |     | |     | |     |  _|_ 1    | |     | |     | |
%e A299778 .    _|_|    _|_|    _|_|    _|_| |_|    _|_|    _|_|    _|_|    _
%e A299778 .   | |     | |     | |     | |         | |     | |     | |     | |
%e A299778 .   | |     | |     | |     |_|_ _     _| |     | |     | |     | |
%e A299778 .   | |     | |     | |    2  |_ _|_ _|  _|     | |     | |     | |
%e A299778 .   | |     | |     |_|_     2    |_ _ _|  0 _ _| |     | |     | |
%e A299778 .   | |     | |    4    |_               7 _|  _ _|0    | |     | |
%e A299778 .   | |     |_|_ _     0  |_ _ _ _        |  _|    _ _ _| |     | |
%e A299778 .   | |    6      |_      |_ _ _ _|_ _ _ _| |  0 _|    _ _|0    | |
%e A299778 .   |_|_ _ _     0  |_   4        |_ _ _ _ _|  _|     |    _ _ _| |
%e A299778 .  8      | |_ _   0  |                     15|      _|   |  _ _ _|
%e A299778 .         |_    |     |_ _ _ _ _ _            |  _ _|  0 _| |      0
%e A299778 .        8  |_  |_    |_ _ _ _ _ _|_ _ _ _ _ _| |    0 _|  _|
%e A299778 .          0  |_ _|  6            |_ _ _ _ _ _ _|  _ _|  _|  0
%e A299778 .            0    |                             28|  _ _|  0
%e A299778 .                 |_ _ _ _ _ _ _ _                | |    0
%e A299778 .                 |_ _ _ _ _ _ _ _|_ _ _ _ _ _ _ _| |
%e A299778 .                8                |_ _ _ _ _ _ _ _ _|
%e A299778 .                                                    31
%e A299778 .
%e A299778 The diagram contains A237590(16) = 27 parts.
%e A299778 For the construction of the spiral see A239660.
%Y A299778 Row sums give A000203.
%Y A299778 Row n has length A003056(n).
%Y A299778 Column k starts in row A000217(k).
%Y A299778 Nonzero terms give A237270.
%Y A299778 The number of nonzero terms in row n is A237271(n).
%Y A299778 Column 1 is A241838.
%Y A299778 The triangle with n rows contain A237590(n) nonzero terms.
%Y A299778 Cf. A296508 (analog for subparts).
%Y A299778 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.
%K A299778 nonn,tabf
%O A299778 1,2
%A A299778 _Omar E. Pol_, Apr 03 2018
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