A141418 - OEIS

%I #18 Mar 31 2021 01:29:03

%S 0,1,1,2,3,3,3,5,6,6,4,7,9,10,10,5,9,12,14,15,15,6,11,15,18,20,21,21,

%T 7,13,18,22,25,27,28,28,8,15,21,26,30,33,35,36,36,9,17,24,30,35,39,42,

%U 44,45,45

%N Triangle read by rows: T(n,k) = k * (2*n - k - 1) / 2, 1 <= k <= n.

%C From _Reinhard Zumkeller_, Aug 04 2014: (Start)

%C n-th row = half of Dynkin diagram weights for the Cartan Groups D_n.

%C n-th row = partial sums of n-th row of A025581. (End)

%D R. N. Cahn, Semi-Simple Lie Algebras and Their Representations, Dover, NY, 2006, ISBN 0-486-44999-8, p. 139.

%H Reinhard Zumkeller, <a href="/A141418/b141418.txt">Rows n = 1..120 of triangle, flattened</a>

%F T(n, K) = k*(2*n - k - 1)/2.

%F Sum_{k=1..n} T(n, k) = 2*binomial(n+1, 3) = A007290(n+1). - _Reinhard Zumkeller_, Aug 04 2014

%e Triangle begins as:

%e   0;

%e   1,  1;

%e   2,  3,  3;

%e   3,  5,  6,  6;

%e   4,  7,  9, 10, 10;

%e   5,  9, 12, 14, 15, 15;

%e   6, 11, 15, 18, 20, 21, 21;

%e   7, 13, 18, 22, 25, 27, 28, 28;

%e   8, 15, 21, 26, 30, 33, 35, 36, 36;

%e   9, 17, 24, 30, 35, 39, 42, 44, 45, 45;

%p A141418:= (n,k)-> k*(2*n-k-1)/2; seq(seq(A141418(n,k), k=1..n), n=1..12); # _G. C. Greubel_, Mar 30 2021

%t T[n_, k_]= k*(2*n-k-1)/2; Table[T[n, k], {n,12}, {k,n}]//Flatten

%o (Haskell)

%o a141418 n k = k * (2 * n - k - 1) `div` 2

%o a141418_row n = a141418_tabl !! (n-1)

%o a141418_tabl = map (scanl1 (+)) a025581_tabl

%o -- _Reinhard Zumkeller_, Aug 04 2014, Nov 18 2012

%o (Magma) [k*(2*n-k-1)/2: k in [1..n], n in [1..12]]; // _G. C. Greubel_, Mar 30 2021

%o (Sage) flatten([[k*(2*n-k-1)/2 for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Mar 30 2021

%Y Cf. A025581, A087401, A141419.

%K nonn,tabl,easy

%O 1,4

%A _Roger L. Bagula_, Aug 05 2008

%E Edited by _Reinhard Zumkeller_, Nov 18 2012