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A133112 Triangular array, read by rows, associated with sums of certain Vandermonde determinants.

%I A133112 #5 Sep 23 2024 05:07:28
%S A133112 1,2,1,3,4,1,4,10,8,1,5,20,35,16,1,6,35,112,126,32,1,7,56,294,672,462,
%T A133112 64,1,8,84,672,2772,4224,1716,128,1,9,120,1386,9504,28314,27456,6435,
%U A133112 256,1,10,165,2640,28314,151008,306735,183040,24310,512,1
%N A133112 Triangular array, read by rows, associated with sums of certain Vandermonde determinants.
%C A133112 Appears to be the same as A073165 read as a triangular array (excluding the first column).
%F A133112 T(n,k)=1/(1!*2! ... *k!)*Sum_{1 <= x_1, ...,x_k <= n} |det V(x_1, ...,x_k)|, where V(x_1, ...,x_k) is the Vandermonde matrix of order k. For example, T(n,2) = 1/2*Sum_{1<=i,j<= n} |i-j|. T(n,k) = 1/(1!*2! ... *k!)*Sum_{1 <= x_1, ...,x_k <= n} |(Product_{1 <= i < j <= k} (x_j - x_i) )|.
%e A133112 Triangle starts:
%e A133112   1;
%e A133112   2 1;
%e A133112   3 4 1;
%e A133112   4 10 8 1;
%e A133112   5 20 35 16 1;
%Y A133112 A000292 (column 2), A040977 (column 3), A133111 (column 4). Cf. A103905.
%K A133112 easy,nonn,tabl
%O A133112 1,2
%A A133112 _Peter Bala_, Sep 18 2007