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A283368 Irregular triangle read by rows: T(n,k) = number of heights for the horizontal elements of the Dyck paths for the symmetric representation of sigma(n) that are listed in the corresponding positions of the triangle of A259176.

%I A283368 #12 Mar 08 2025 18:38:51
%S A283368 1,2,3,2,4,3,5,3,6,5,4,7,5,4,8,6,5,9,7,5,10,8,7,6,11,8,7,6,12,10,9,7,
%T A283368 13,10,9,7,14,11,9,8,15,12,11,10,8,16,13,12,11,9,17,13,12,11,9,18,15,
%U A283368 13,12,10,19,15,13,12,10,20,16,15,13,11,21,17,16,15,14,11
%N A283368 Irregular triangle read by rows: T(n,k) = number of heights for the horizontal elements of the Dyck paths for the symmetric representation of sigma(n) that are listed in the corresponding positions of the triangle of A259176.
%C A283368 The dot product of the n-th row of this triangle and the n-th row of triangle A259176 equals A024916(n), the sum of all divisors of numbers 1 through n (true for all n <= 20000); the value is the sum of the rectangles between the x-axis and the horizontal legs of the symmetric representation of sigma(n). This is the companion computation to A283367.
%F A283368 T(n,k) = n - Sum_{i=1..k-1} f(n, 2*i) where f is defined in A237593.
%F A283368 A024916(n) = Sum_{i=1..row(n)} T(n,i)*S(n,i) where S(n,i) refers to the triangle of A259176 and row(n) = floor((sqrt(8*n+1)-1)/2).
%e A283368 The first horizontal leg of the symmetric representation of sigma(15) is at y-coordinate 15 and has length 8, and row 15 has 5 entries so that T(15,1) = 15 and T(15,5) = 8.
%e A283368 The first 16 rows of the irregular triangle:
%e A283368    1
%e A283368    2
%e A283368    3    2
%e A283368    4    3
%e A283368    5    3
%e A283368    6    5    4
%e A283368    7    5    4
%e A283368    8    6    5
%e A283368    9    7    5
%e A283368   10    8    7    6
%e A283368   11    8    7    6
%e A283368   12   10    9    7
%e A283368   13   10    9    7
%e A283368   14   11    9    8
%e A283368   15   12   11   10    8
%e A283368   16   13   12   11    9
%t A283368 (* function f[n,k] and its support functions are defined in A237593 *)
%t A283368 a283368[n_, k_] := n - Sum[f[n, 2i], {i, k-1}]
%t A283368 TableForm[Table[a283368[n, k], {n, 1, 16}, {k, 1, row[n]}]] (* triangle *)
%t A283368 Flatten[Table[a283368[n, k], {n, 1, 21}, {k, 1, row[n]}]] (* sequence data *)
%Y A283368 Cf. A024916, A237593, A259176, A259177, A283367.
%K A283368 nonn,tabf
%O A283368 1,2
%A A283368 _Hartmut F. W. Hoft_, Mar 06 2017