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A360022 Triangle read by rows: T(n,k) is the sum of the widths of the k-th diagonals of the symmetric representation of sigma(n).

%I A360022 #37 Feb 05 2023 13:49:16
%S A360022 1,1,2,0,2,2,1,2,2,2,0,0,2,2,2,2,2,2,2,2,2,0,0,0,2,2,2,2,1,2,2,2,2,2,
%T A360022 2,2,1,2,0,0,2,2,2,2,2,0,2,2,2,2,2,2,2,2,2,0,0,0,0,0,2,2,2,2,2,2,2,4,
%U A360022 4,2,2,2,2,2,2,2,2,2,0,0,0,0,0,0,2,2,2,2,2,2,2,0,0,2,2,2,2,2,2,2,2,2,2,2,2
%N A360022 Triangle read by rows: T(n,k) is the sum of the widths of the k-th diagonals of the symmetric representation of sigma(n).
%C A360022 The main diagonal of the diagram called "symmetric representation of sigma(n)" is its axis of symmetry. In this case it is also the first diagonal of the diagram. The second diagonals are the two diagonals that are adjacent to the main diagonal. The third diagonals are the two diagonals that are adjacent to the second diagonals. And so on.
%C A360022 If and only if n is a power of 2 (A000079) then row n lists the first n terms of A040000 (the same sequence as the right border of the triangle).
%C A360022 If and only if n is an odd prime (A065091) then row n lists (n - 1)/2 zeros together with 1 + (n - 1)/2 2's.
%C A360022 If and only if n is an even perfect number (Cf. A000396) then row n lists n 2's (the first n terms of A007395).
%C A360022 For further information about the mentioned "widths" see A249351.
%H A360022 Omar E. Pol, <a href="/A067742/a067742.jpg">Illustration of initial terms of the column 1 = A067742</a>
%H A360022 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F A360022 T(n,1) = A067742(n) = A249351(n,n).
%F A360022 T(n,k) = 2*A249351(n,n+k-1), if 1 < k <= n.
%e A360022 Triangle begins (rows: 1..16):
%e A360022   1;
%e A360022   1, 2;
%e A360022   0, 2, 2;
%e A360022   1, 2, 2, 2;
%e A360022   0, 0, 2, 2, 2;
%e A360022   2, 2, 2, 2, 2, 2;
%e A360022   0, 0, 0, 2, 2, 2, 2;
%e A360022   1, 2, 2, 2, 2, 2, 2, 2;
%e A360022   1, 2, 0, 0, 2, 2, 2, 2, 2;
%e A360022   0, 2, 2, 2, 2, 2, 2, 2, 2, 2;
%e A360022   0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 2;
%e A360022   2, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2;
%e A360022   0, 0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 2, 2;
%e A360022   0, 0, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2;
%e A360022   2, 2, 2, 2, 0, 0, 0, 2, 2, 2, 2, 2, 2, 2, 2;
%e A360022   1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2;
%e A360022   ...
%Y A360022 Row sums give A000203.
%Y A360022 Column 1 gives A067742.
%Y A360022 Right border gives A040000.
%Y A360022 Cf. A000079, A000396, A007395, A065091, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A245092, A249351, A250068, A250070, A262626.
%K A360022 nonn,tabl
%O A360022 1,3
%A A360022 _Omar E. Pol_, Jan 22 2023