OFFSET
1,4
COMMENTS
From Reinhard Zumkeller, Aug 04 2014: (Start)
n-th row = half of Dynkin diagram weights for the Cartan Groups D_n.
n-th row = partial sums of n-th row of A025581. (End)
REFERENCES
R. N. Cahn, Semi-Simple Lie Algebras and Their Representations, Dover, NY, 2006, ISBN 0-486-44999-8, p. 139.
LINKS
Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened
FORMULA
T(n, K) = k*(2*n - k - 1)/2.
Sum_{k=1..n} T(n, k) = 2*binomial(n+1, 3) = A007290(n+1). - Reinhard Zumkeller, Aug 04 2014
EXAMPLE
Triangle begins as:
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;
7, 13, 18, 22, 25, 27, 28, 28;
8, 15, 21, 26, 30, 33, 35, 36, 36;
9, 17, 24, 30, 35, 39, 42, 44, 45, 45;
MAPLE
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
MATHEMATICA
T[n_, k_]= k*(2*n-k-1)/2; Table[T[n, k], {n, 12}, {k, n}]//Flatten
PROG
(Haskell)
a141418 n k = k * (2 * n - k - 1) `div` 2
a141418_row n = a141418_tabl !! (n-1)
a141418_tabl = map (scanl1 (+)) a025581_tabl
-- Reinhard Zumkeller, Aug 04 2014, Nov 18 2012
(Magma) [k*(2*n-k-1)/2: k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 30 2021
(Sage) flatten([[k*(2*n-k-1)/2 for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Mar 30 2021
CROSSREFS
KEYWORD
AUTHOR
Roger L. Bagula, Aug 05 2008
EXTENSIONS
Edited by Reinhard Zumkeller, Nov 18 2012
STATUS
approved