A208656 Triangle T(n, k) = n*C(n,k) - C(n-1,k-1), 1 <= k <= n, read by rows.

0, 3, 1, 8, 7, 2, 15, 21, 13, 3, 24, 46, 44, 21, 4, 35, 85, 110, 80, 31, 5, 48, 141, 230, 225, 132, 43, 6, 63, 217, 427, 525, 413, 203, 57, 7, 80, 316, 728, 1078, 1064, 700, 296, 73, 8, 99, 441, 1164, 2016, 2394, 1974, 1116, 414, 91, 9, 120, 595, 1770

Offset: 1

Author

Clark Kimberling, Mar 01 2012

Keywords

• nonn
• tabl

Comments

Mirror of A208657.

col 1: A005563

col 2: A127736

top edge: A000027

Examples

			First five rows:
0
3....1
8....7....2
15...21...13...3
24...46...44...21...4
		

Crossrefs

Cf. A208657, A208658.

Programs

Mathematica

z = 12;
f[n_, k_] := n*Binomial[n, k] - Binomial[n - 1, k - 1]
t = Table[f[n, k], {n, 1, z}, {k, 1, n}];
TableForm[t] (* A208656 as a triangle *)
Flatten[t]   (* A208656 as a sequence *)
r = Table[f[n, k], {n, 1, z}, {k, n, 1, -1}];
TableForm[r] (* A208657 as a triangle *)
Flatten[r]   (* A208657 as a sequence *)
Table[Sum[f[n, k], {k, 1, n}], {n, 1, 3 z}](* A208658 *)

Extensions

Definition amended by Georg Fischer, Feb 01 2022