cp's OEIS Frontend

A346873 Triangle read by rows in which row n lists the row A000217(n) of A237591, n >= 1.

%I A346873 #51 Feb 07 2023 06:01:51
%S A346873 1,2,1,4,1,1,6,2,1,1,8,3,2,1,1,11,4,3,1,1,1,15,5,3,2,1,1,1,19,6,4,2,2,
%T A346873 1,1,1,23,8,5,2,2,2,1,1,1,28,10,5,3,3,2,1,1,1,1,34,11,6,4,3,2,2,1,1,1,
%U A346873 1,40,13,7,5,3,2,2,2,1,1,1,1,46,16,8,5,4,2,3
%N A346873 Triangle read by rows in which row n lists the row A000217(n) of A237591, n >= 1.
%C A346873 The characteristic shape of the symmetric representation of sigma(A000217(n)) consists in that in the main diagonal of the diagram the smallest Dyck path has a valley and the largest Dyck path has a peak, or vice versa, the smallest Dyck path has a peak and the largest Dyck path has valley.
%C A346873 So knowing this characteristic shape we can know if a number is a triangular number (or not) just by looking at the diagram, even ignoring the concept of triangular number.
%C A346873 Therefore we can see a geometric pattern of the distribution of the triangular numbers in the stepped pyramid described in A245092.
%C A346873 T(n,k) is also the length of the k-th line segment of the largest Dyck path of the symmetric representation of sigma(A000217(n)), from the border to the center, hence the sum of the n-th row of triangle is equal to A000217(n).
%C A346873 T(n,k) is also the difference between the total number of partitions of all positive integers <= n-th triangular number into exactly k consecutive parts, and the total number of partitions of all positive integers <= n-th triangular number into exactly k + 1 consecutive parts.
%F A346873 T(n,k) = A237591(A000217(n),k). - _Omar E. Pol_, Feb 06 2023
%e A346873 Triangle begins:
%e A346873    1;
%e A346873    2,  1;
%e A346873    4,  1, 1;
%e A346873    6,  2, 1, 1;
%e A346873    8,  3, 2, 1, 1;
%e A346873   11,  4, 3, 1, 1, 1;
%e A346873   15,  5, 3, 2, 1, 1, 1;
%e A346873   19,  6, 4, 2, 2, 1, 1, 1;
%e A346873   23,  8, 5, 2, 2, 2, 1, 1, 1;
%e A346873   28, 10, 5, 3, 3, 2, 1, 1, 1, 1;
%e A346873   34, 11, 6, 4, 3, 2, 2, 1, 1, 1, 1;
%e A346873   40, 13, 7, 5, 3, 2, 2, 2, 1, 1, 1, 1;
%e A346873   46, 16, 8, 5, 4, 2, 3, 1, 2, 1, 1, 1, 1;
%e A346873 ...
%e A346873 Illustration of initial terms:
%e A346873 Column T gives the triangular numbers (A000217).
%e A346873 Column S gives A074285, the sum of the divisors of the triangular numbers which equals the area (and the number of cells) of the associated diagram.
%e A346873 -------------------------------------------------------------------------
%e A346873   n    T    S   Diagram
%e A346873 -------------------------------------------------------------------------
%e A346873                  _   _     _       _         _           _             _
%e A346873   1    1    1   |_| | |   | |     | |       | |         | |           | |
%e A346873                1 _ _|_|   | |     | |       | |         | |           | |
%e A346873   2    3    4   |_ _|  _ _| |     | |       | |         | |           | |
%e A346873                   2  1|    _|     | |       | |         | |           | |
%e A346873                  _ _ _|  _|    _ _| |       | |         | |           | |
%e A346873   3    6   12   |_ _ _ _| 1   |  _ _|       | |         | |           | |
%e A346873                     4    1 _ _|_|           | |         | |           | |
%e A346873                           |  _|1       _ _ _|_|         | |           | |
%e A346873                  _ _ _ _ _| | 1    _ _| |               | |           | |
%e A346873   4   10   18   |_ _ _ _ _ _|2    |    _|               | |           | |
%e A346873                       6          _|  _|          _ _ _ _|_|           | |
%e A346873                                 |_ _|1 1        | |                   | |
%e A346873                                 | 2            _| |                   | |
%e A346873                  _ _ _ _ _ _ _ _|4            |  _|          _ _ _ _ _| |
%e A346873   3   15   24   |_ _ _ _ _ _ _ _|          _ _|_|           |  _ _ _ _ _|
%e A346873                         8              _ _|  _|1            | |
%e A346873                                       |_ _ _|1 1         _ _| |
%e A346873                                       |  3           _ _|  _ _|
%e A346873                                       |4            |    _|
%e A346873                  _ _ _ _ _ _ _ _ _ _ _|            _|  _|
%e A346873   4   21   32   |_ _ _ _ _ _ _ _ _ _ _|      _ _ _|  _|1 1
%e A346873                           11                |  _ _ _|2
%e A346873                                             | |  3
%e A346873                                             | |
%e A346873                                             | |5
%e A346873                  _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
%e A346873   5   28   56   |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
%e A346873                               15
%e A346873 .
%Y A346873 Row sums give A000217, n >= 1.
%Y A346873 Column 1 gives A039823.
%Y A346873 For the characteristic shape of sigma(A000040(n)) see A346871.
%Y A346873 For the characteristic shape of sigma(A000079(n)) see A346872.
%Y A346873 For the visualization of Mersenne numbers A000225 see A346874.
%Y A346873 For the characteristic shape of sigma(A000384(n)) see A346875.
%Y A346873 For the characteristic shape of sigma(A000396(n)) see A346876.
%Y A346873 For the characteristic shape of sigma(A008588(n)) see A224613.
%Y A346873 Cf. A000203, A074285, A237591, A237593, A245092, A249351, A262626.
%K A346873 nonn,tabl
%O A346873 1,2
%A A346873 _Omar E. Pol_, Aug 06 2021
%E A346873 Name corrected by _Omar E. Pol_, Feb 06 2023