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

0,5

Comments

Reverse of triangle A104878.

Links

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

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).

Example

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

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 *)

Prog

(Magma) [(&+[ (n-k)^(k-j): j in [0..k]]): k in [0..n], n in [0..12]]; // 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

Crossrefs

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

Sequence in context: A327913 A028657 A053534 * A171699 A104878 A196863

Adjacent sequences: A104878 A104879 A104880 * A104882 A104883 A104884

Keyword

easy,nonn,tabl

Author

Paul Barry, Mar 28 2005

Status

approved