%I A141387
%S A141387 0,1,1,2,4,2,3,7,7,3,4,10,12,10,4,5,13,17,17,13,5,6,16,22,24,22,16,6,7,
%T A141387 19,27,31,31,27,19,7,8,22,32,38,40,38,32,22,8,9,25,37,45,49,49,45,37,25,
%U A141387 9,10,28,42,52,58,60,58,52,42,28,10
%N A141387 Overlapping weights of Dynkin diagrams for the A_n Cartan group: T(n,
               m)=(n-m)*(m+1)+(n-(m-1))*m=n+2*m*(n-m).
%C A141387 Row sums are: {0, 2, 8, 20, 40, 70, 112, 168, 240, 330, 440, ...}.
%C A141387 The sequence based on a dual overlap of A003991 or the weights of Cartan 
               A_n
%C A141387 Dynkin diagrams ( Cahn above) :
%C A141387 t(n,m)=(n-m)*(m+1)->(n - m)*(m + 1) + (n - (m - 1))*m.
%C A141387 I call these neo-combinations because they are very like the symmetrical 
               commutative/ Abelian group based combinations,
%C A141387 but produced by a Cartan Lie type algebra.
%C A141387 t(n,m)=Floor[T(n,m)/n]:
%C A141387 t[n_, m_] = If[n == m == 0, 0, Floor[(n + 2* m *(-m + n))/n]]
%C A141387 is very like the binomial in form.
%D A141387 R. N. Cahn, Semi-Simple Lie Algebras and Their Representations, Dover, 
               NY, 2006, ISBN 0-486-44999-8, p. 139.
%F A141387 T(n,m)=n+2*m*(n-m).
%e A141387 {0},
%e A141387 {1, 1},
%e A141387 {2, 4, 2},
%e A141387 {3, 7, 7, 3},
%e A141387 {4, 10, 12, 10, 4},
%e A141387 {5, 13, 17, 17, 13, 5},
%e A141387 {6, 16, 22, 24, 22, 16, 6},
%e A141387 {7, 19, 27, 31, 31, 27, 19, 7},
%e A141387 {8, 22, 32, 38, 40, 38, 32, 22, 8},
%e A141387 {9, 25, 37, 45, 49, 49, 45, 37, 25, 9},
%e A141387 {10, 28, 42, 52, 58, 60, 58, 52, 42, 28, 10}
%t A141387 Clear[T, n, m, a]; T[n_, m_] = n + 2* m *(-m + n); a = Table[Table[T[n, 
               m], {m, 0, n}], {n, 0, 10}]; Flatten[a]
%Y A141387 Cf. A003991.
%Y A141387 Sequence in context: A134447 A093056 A151849 this_sequence A134400 A016095 
               A165464
%Y A141387 Adjacent sequences: A141384 A141385 A141386 this_sequence A141388 A141389 
               A141390
%K A141387 nonn,uned,tabl
%O A141387 1,4
%A A141387 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Aug 03 2008