A199011 Triangle T(n,k), read by rows, given by (1,1,-1,1,0,0,0,0,0,0,0,...) DELTA (0,1,0,0,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

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

Offset: 0

Author

Philippe Deléham, Nov 01 2011

Keywords

• easy
• nonn
• tabl

Comments

Mirror image of triangle in A198321.

Variant of A074909, A135278.

Examples

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

Crossrefs

Cf. A007318, A074909, A135278, A198321

Formula

T(n,k)=binomial(n,k+1).

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

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

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