A049780 Array T, read by descending antidiagonals: T(n, k) = k*(2*n + k + 1)/2 for n, k >= 0.

0, 1, 0, 3, 2, 0, 6, 5, 3, 0, 10, 9, 7, 4, 0, 15, 14, 12, 9, 5, 0, 21, 20, 18, 15, 11, 6, 0, 28, 27, 25, 22, 18, 13, 7, 0, 36, 35, 33, 30, 26, 21, 15, 8, 0, 45, 44, 42, 39, 35, 30, 24, 17, 9, 0, 55, 54, 52, 49, 45, 40, 34, 27, 19, 10, 0, 66, 65, 63, 60, 56, 51, 45, 38, 30, 21, 11, 0

Offset

0,4

Comments

Triangle S(n,k) = T(k, n-k), read by rows, is given by S(n,k) = A000217(n) - A000217(k) for n >= 0 and 0 <= k <= n. - Philippe Deléham, Mar 07 2013 [Edited by Petros Hadjicostas, Nov 20 2019]

Links

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Formula

T(n,k) = Sum_{j=1..k} (n+j) = k*(2*n + k + 1)/2.

S(n,k) = n*(n+1)/2 - k*(k+1)/2 for n >= 0 and 0 <= k <= n - Philippe Deléham, Mar 07 2013 [Edited by Petros Hadjicostas, Nov 20 2019]

From Stefano Spezia, Dec 13 2019: (Start)

G.f. for T(n,k): y*(1 - x*y)/((1 - x)^2*(1 - y)^3).

E.g.f. for T(n,k): (1/2)*exp(x+y)*y*(2 + 2*x + y). (End)

G.f. for S(n,k): x*(1 - x^2*y)/((1 - x*y)^2*(1 - x)^3). - Petros Hadjicostas, Dec 14 2019

Example

From Petros Hadjicostas, Nov 20 2019: (Start)
Rectangular array T(n,k) (with rows n >= 0 and columns k >= 0) begins as follows:
  0, 1,  3,  6, 10, 15, 21, 28, 36, 45,  55, ...
  0, 2,  5,  9, 14, 20, 27, 35, 44, 54,  65, ...
  0, 3,  7, 12, 18, 25, 33, 42, 52, 63,  75, ...
  0, 4,  9, 15, 22, 30, 39, 49, 60, 72,  85, ...
  0, 5, 11, 18, 26, 35, 45, 56, 68, 81,  95, ...
  0, 6, 13, 21, 30, 40, 51, 63, 76, 90, 105, ...
  ...
From Philippe Deléham, Mar 07 2013: (Start)
Triangle S(n, k) (with rows n >= 0 and columns k >= 0) begins as follows:
   0;
   1,  0;
   3,  2,  0;
   6,  5,  3,  0;
  10,  9,  7,  4,  0;
  15, 14, 12,  9,  5,  0;
  21, 20, 18, 15, 11,  6,  0;
  28, 27, 25, 22, 18, 13,  7, 0;
  36, 35, 33, 30, 26, 21, 15, 8, 0;
  ... (End)

Maple

seq(seq( (n-k)*(n+k+1)/2, k=0..n), n=0..15); # G. C. Greubel, Dec 12 2019

Mathematica

Table[(n-k)*(n+k+1)/2, {n, 0, 15}, {k, 0, n}]//Flatten (* G. C. Greubel, Dec 12 2019 *)

Prog

(PARI) T(n, k) = k*(2*n+k+1)/2;
for(n=0, 15, for(k=0, n, print1(T(k, n-k), ", "))) \\ G. C. Greubel, Dec 12 2019
(Magma) [(n-k)*(n+k+1)/2: k in [0..n], n in [0..15]]; // G. C. Greubel, Dec 12 2019
(Sage) [[(n-k)*(n+k+1)/2 for k in (0..n)] for n in (0..15)] # G. C. Greubel, Dec 12 2019
(GAP) Flat(List([0..15], n-> List([0..n], k-> (n-k)*(n+k+1)/2 ))); # G. C. Greubel, Dec 12 2019

Crossrefs

Diagonal sums are in A000330. See also A049777 (triangle without the zeros).

Sequence in context: A114376 A216395 A058096 * A159584 A257653 A246834

Adjacent sequences: A049777 A049778 A049779 * A049781 A049782 A049783

Keyword

nonn,tabl

Author

Clark Kimberling

Status

approved