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A283367 Irregular triangle read by rows: T(n,k) = number of horizontal positions for the vertical legs of the Dyck paths for the symmetric representation of sigma(n) that are listed in the corresponding positions of the triangle of A259177.

%I A283367 #14 Mar 08 2025 17:17:36
%S A283367 1,2,2,3,3,4,3,5,4,5,6,4,5,7,5,6,8,5,7,9,6,7,8,10,6,7,8,11,7,9,10,12,
%T A283367 7,9,10,13,8,9,11,14,8,10,11,12,15,9,11,12,13,16,9,11,12,13,17,10,12,
%U A283367 13,15,18,10,12,13,15,19,11,13,15,16,20,11,14,15,16,17,21
%N A283367 Irregular triangle read by rows: T(n,k) = number of horizontal positions for the vertical legs of the Dyck paths for the symmetric representation of sigma(n) that are listed in the corresponding positions of the triangle of A259177.
%C A283367 The dot product of the n-th row of this triangle and the n-th row of triangle A259177 equals A024916(n), the sum of all divisors of numbers 1 through n (true for all n <= 20000); the value of a(n) is the sum of the rectangles between the y-axis and the vertical legs of the symmetric representation of sigma(n). This is the companion computation to A283368.
%F A283367 T(n,k) = Sum_{i=1..k} f(n, 2*i-1) where f is defined in A237593.
%F A283367 A024916(n) = Sum_{i=1..row(n)} T(n,i)*S(n,i) where S(n,i) refers to the triangle of A259177 and row(n) = floor((sqrt(8*n+1)-1)/2).
%e A283367 The first vertical leg of the symmetric representation of sigma(15) is at x-coordinate 8 and has length 3, and row 15 has 5 entries so that T(15,1) = 8 and T(15,5) = 15.
%e A283367 The first 16 rows of the irregular triangle:
%e A283367    1:   1
%e A283367    2:   2
%e A283367    3:   2    3
%e A283367    4:   3    4
%e A283367    5:   3    5
%e A283367    6:   4    5    6
%e A283367    7:   4    5    7
%e A283367    8:   5    6    8
%e A283367    9:   5    7    9
%e A283367   10:   6    7    8   10
%e A283367   11:   6    7    8   11
%e A283367   12:   7    9   10   12
%e A283367   13:   7    9   10   13
%e A283367   14:   8    9   11   14
%e A283367   15:   8   10   11   12   15
%e A283367   16:   9   11   12   13   16
%t A283367 (* function f[n,k] and its support functions are defined in A237593 *)
%t A283367 a283367[n_, k_] := Sum[f[n, 2*i-1], {i, k}]
%t A283367 TableForm[Table[a283367[n, k], {n, 1, 16}, {k, 1, row[n]}]] (* triangle *)
%t A283367 Flatten[Table[a283367[n, k], {n, 1, 21}, {k, 1, row[n]}]] (* sequence data *)
%Y A283367 Cf. A024916, A237593, A259176, A259177, A283368.
%K A283367 nonn,tabf
%O A283367 1,2
%A A283367 _Hartmut F. W. Hoft_, Mar 06 2017