A198321 Triangle read by rows: T(n, k) = binomial(n, k-1) for 1 <= k <= n, and T(n, 0) = 0^n.

1, 0, 1, 0, 1, 2, 0, 1, 3, 3, 0, 1, 4, 6, 4, 0, 1, 5, 10, 10, 5, 0, 1, 6, 15, 20, 15, 6, 0, 1, 7, 21, 35, 35, 21, 7, 0, 1, 8, 28, 56, 70, 56, 28, 8, 0, 1, 9, 36, 84, 126, 126, 84, 36, 9, 0, 1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 0, 1, 11, 55, 165, 330

Offset: 0

Author

Philippe Deléham, Nov 01 2011

Keywords

• easy
• nonn
• tabl

Examples

			Triangle begins :
1
0, 1
0, 1, 2
0, 1, 3, 3
0, 1, 4, 6, 4
0, 1, 5, 10, 10, 5
0, 1, 6, 15, 20, 15, 6
		

Links

• Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of triangle, flattened).

Crossrefs

Variant of A074909, A135278.

Cf. A007318.

Programs

Mathematica

A198321[n_, k_] := If[k == 0, Boole[n == 0], Binomial[n, k - 1]];
Table[A198321[n, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Oct 23 2024 *)

Formula

T(n, k) is given by (0,1,0,0,0,0,0,0,0,0,0,...) DELTA (1,1,-1,1,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

Sum_{k=0..n} T(n, k)*x^k = x*((x+1)^n - x^n) for n > 0.

G.f.: (1 - (1+y)*x + y*(1+y)*x^2)/((1 - (1+y)*x)*(1-y*x)).

T(n, k) = T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-1) - T(n-2,k-2), T(0,0) = 1, T(1,0) = 0, T(1,1) = 1, T(2,0) = 0, T(2,1) = 1, T(2,2) = 2, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Feb 12 2014

Extensions

New name using a formula of the author by Peter Luschny, Oct 23 2024