A198321 Triangle T(n,k), read by rows, 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.

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

0,6

Comments

Variant of A074909, A135278.

Links

Table of n, a(n) for n=0..71.

Formula

T(n,0)=0^n, T(n,k)=binomial(n,k-1) for 1<=k<=n.

Sum_{0<=k<=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

Example

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

Crossrefs

Cf. A007318, A074909, A135278.

Sequence in context: A089112 A155584 A139600 * A325003 A166278 A365515

Adjacent sequences: A198318 A198319 A198320 * A198322 A198323 A198324

Keyword

easy,nonn,tabl

Author

Philippe Deléham, Nov 01 2011

Status

approved