A141418 Triangle read by rows: T(n,k) = k * (2*n - k - 1) / 2, 1 <= k <= n.

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

Offset: 1

Author

Roger L. Bagula, Aug 05 2008

Keywords

• nonn
• tabl
• easy

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)

Examples

			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;
		

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

Crossrefs

Cf. A025581, A087401, A141419.

Programs

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

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

Sage

flatten([[k*(2*n-k-1)/2 for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Mar 30 2021

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

Extensions

Edited by Reinhard Zumkeller, Nov 18 2012