A104881 Triangle T(n,k) = Sum_{j=0..k} (n-k)^(k-j), read by rows.

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 7, 4, 1, 1, 5, 13, 15, 5, 1, 1, 6, 21, 40, 31, 6, 1, 1, 7, 31, 85, 121, 63, 7, 1, 1, 8, 43, 156, 341, 364, 127, 8, 1, 1, 9, 57, 259, 781, 1365, 1093, 255, 9, 1, 1, 10, 73, 400, 1555, 3906, 5461, 3280, 511, 10, 1, 1, 11, 91, 585, 2801, 9331, 19531, 21845, 9841, 1023, 11, 1

Offset: 0

Author

Paul Barry, Mar 28 2005

Keywords

• easy
• nonn
• tabl

Comments

Reverse of triangle A104878.

Examples

			Triangle begins as:
  1;
  1, 1;
  1, 2,  1;
  1, 3,  3,  1;
  1, 4,  7,  4, 1;
  1, 5, 13, 15, 5, 1;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Crossrefs

Cf. A104878, A104879 (row sums), A104882 (diagonal sums).

Programs

Magma

[(&+[ (n-k)^(k-j): j in [0..k]]): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 15 2021

Mathematica

T[n_, k_]:= If[k==n, 1, Sum[(n-k)^(k-j), {j,0,k}]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 15 2021 *)

Sage

flatten([[sum((n-k)^(k-j) for j in (0..k)) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 15 2021

Formula

T(n, k) = Sum_{j=0..k} (n-k)^(k-j).

Sum_{k=0..n} T(n, k) = A104879(n).

Sum_{k=0..floor(n/2)} T(k, n-k) = A104882(n).