A258197 Arithmetic derivative of Pascal's triangle.

0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 4, 5, 4, 0, 0, 1, 7, 7, 1, 0, 0, 5, 8, 24, 8, 5, 0, 0, 1, 10, 12, 12, 10, 1, 0, 0, 12, 32, 92, 59, 92, 32, 12, 0, 0, 6, 60, 124, 165, 165, 124, 60, 6, 0, 0, 7, 39, 244, 247, 456, 247, 244, 39, 7, 0, 0, 1, 16, 103, 371, 493, 493, 371, 103, 16, 1, 0

Offset: 0

Author

Reinhard Zumkeller, May 26 2015

Keywords

• nonn
• tabl

Comments

A258318(n) = number of distinct terms up to row n.

Examples

			.  First 10 rows:                     |  Pascal's triangle:
.  0:  0                              |  1
.  1:  0,0                            |  1,1
.  2:  0,1,0                          |  1,2,1
.  3:  0,1,1,0                        |  1,3,3,1
.  4:  0,4,5,4,0                      |  1,4,6,4,1
.  5:  0,1,7,7,1,0                    |  1,5,10,10,5,1
.  6:  0,5,8,24,8,5,0                 |  1,6,15,20,15,6,1
.  7:  0,1,10,12,12,10,1,0            |  1,7,21,35,35,21,7,1
.  8:  0,12,32,92,59,92,32,12,0       |  1,8,28,56,70,56,28,8,1
.  9:  0,6,60,124,165,165,124,60,6,0  |  1,9,36,84,126,126,84,36,9,1 .
		

Links

• Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened

Crossrefs

Cf. A003415, A007318, A258290 (central terms), A258317 (row sums), A068312, A258318.

Programs

Haskell

a258197 n k = a258197_tabl !! n !! k
a258197_row n = a258197_tabl !! n
a258197_tabl = map (map a003415) a007318_tabl

Mathematica

ad[n_] := n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); ad[0] = ad[1] = 0; Table[ad[Binomial[n, k]], {n, 0, 11}, {k, 0, n}] // Flatten (* Amiram Eldar, Apr 13 2025 *)

Formula

T(n,k) = A003415(A007318(n,k)), 0 <= k <= n.

For n > 0: T(n,1) = A003415(n).

For n > 1: T(n,2) = A068312(n-1).