|
%I #12 Feb 04 2023 20:48:00
%S 1,2,4,2,3,6,9,6,3,4,8,12,16,12,8,4,5,10,15,20,25,20,15,10,5,6,12,18,
%T 24,30,36,30,24,18,12,6,7,14,21,28,35,42,49,42,35,28,21,14,7,8,16,24,
%U 32,40,48,56,64,56,48,40,32,24,16,8,9,18,27,36,45,54,63,72,81
%N Triangular array T read by rows related to the multiplication table.
%F T(n, k) = T(n, 2*n-k) = n*k for 1<=k<=n .
%F Sum_{k=1..2*n-1} T(n, k) = n^3.
%F Sum_{k=1..2*n-1} T(n, k)^2 = n^3*(2*n^2 + 1)/3 = A272125(n).
%F T(n, k) = n * A004737(n,k).
%e Table T(n, k) , n>=1 , 1<=k<=2*n-1.
%e n = 1 : 1 ;
%e n = 2 : 2, 4, 2 ;
%e n = 3 : 3, 6, 9, 6, 3 ;
%e n = 4 : 4, 8, 12, 16, 12, 8, 4 ;
%e n = 5 : 5, 10, 15, 20, 25, 20, 15, 10, 5 ;
%e n = 6 : 6, 12, 18, 24, 30, 36, 30, 24, 18, 12, 6 ;
%e n = 7 : 7, 14, 21, 28, 35, 42, 49, 42, 35, 28, 21, 14, 7 ;
%e n = 8 : 8, 16, 24, 32, 40, 48, 56, 64, 56, 48, 40, 32, 24, 16, 8 ;
%e ...
%p T:= (n, k)-> n*min(k, 2*n-k):
%p seq(seq(T(n,k), k=1..2*n-1), n=1..10); # _Alois P. Heinz_, Feb 04 2023
%Y Cf. A000290 (central terms), A000578 (row sums), A060747 (row lengths).
%Y Cf. A003991, A004737, A075362, A272125 .
%K nonn,tabf
%O 1,2
%A _Philippe Deléham_, Feb 04 2023
|
