A091186 Triangle read by rows, in which n-th row gives expansion of x^n/((1-x)(1-x-x^2)^n).

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 7, 8, 4, 1, 1, 12, 18, 13, 5, 1, 1, 20, 38, 35, 19, 6, 1, 1, 33, 76, 86, 59, 26, 7, 1, 1, 54, 147, 197, 164, 91, 34, 8, 1, 1, 88, 277, 430, 420, 281, 132, 43, 9, 1, 1, 143, 512, 904, 1014, 792, 447, 183, 53, 10, 1, 1, 232, 932, 1846, 2338, 2087, 1371

Offset: 0

Author

Paul Barry, Dec 25 2003

Keywords

• easy
• nonn
• tabl

Comments

Riordan array (1/(1-x),x/(1-x-x^2)). - Paul Barry, Sep 13 2006

Examples

			Rows begin {1},{1,1},{1,2,1},{1,4,3,1}...
		

Crossrefs

Row sums are A024537. Diagonal sums are A005578. Second column is A000071. Third column is A006478.

Essentially the vertical partial sums of triangle A037027.

Formula

G.f.: (1-y-y^2) / [(1-y(1+y+z))(1-y)].

Number triangle T(n,k)=sum{j=0..n-k, sum{i=0..n-k-j, C(k+j-1,j)C(j,n-k-i-j)}}; - Paul Barry, Sep 13 2006

T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k-1) - T(n-3,k), T(0,0) = T(1,0) = T(1,1) = T(2,0) = T(2,2) = 1, T(2,1) = 2, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Jan 20 2014