%I #9 Jan 10 2018 20:32:27
%S 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 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 13,15,18,10,12,13,15,19,11,13,15,16,20,11,14,15,16,17,21
%N 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 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 T(n,k) = sum_{i=1..k} f(n, 2*i-1) where f is defined in A237593.
%F 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 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 The first 16 rows of the irregular triangle:
%e 1: 1
%e 2: 2
%e 3: 2 3
%e 4: 3 4
%e 5: 3 5
%e 6: 4 5 6
%e 7: 4 5 7
%e 8: 5 6 8
%e 9: 5 7 9
%e 10: 6 7 8 10
%e 11: 6 7 8 11
%e 12: 7 9 10 12
%e 13: 7 9 10 13
%e 14: 8 9 11 14
%e 15: 8 10 11 12 15
%e 16: 9 11 12 13 16
%t (* function f[n,k] and its support functions are defined in A237593 *)
%t a283367[n_, k_] := Sum[f[n, 2*i-1], {i, k}]
%t TableForm[Table[a283367[n, k], {n, 1, 16}, {k, 1, row[n]}]] (* triangle *)
%t Flatten[Table[a283367[n, k], {n, 1, 21}, {k, 1, row[n]}]] (* sequence data *)
%Y Cf. A024916, A237593, A259176, A259177, A283368.
%K nonn,tabf
%O 1,2
%A _Hartmut F. W. Hoft_, Mar 06 2017