A210636 Riordan array ((1-x)/(1-2*x-x^2), x*(1+x)/(1-2*x-x^2)).

1, 1, 1, 3, 4, 1, 7, 13, 7, 1, 17, 40, 32, 10, 1, 41, 117, 124, 60, 13, 1, 99, 332, 437, 286, 97, 16, 1, 239, 921, 1447, 1193, 553, 143, 19, 1, 577, 2512, 4584, 4556, 2682, 952, 198, 22, 1, 1393, 6761, 14048, 16336, 11666, 5282, 1510, 262, 25, 1

Offset: 0

Author

Philippe Deléham, Mar 26 2012

Keywords

• easy
• nonn
• tabl

Comments

Triangle T(n,k), 0<=k<=n, read by rows, given by (1, 2, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Product of A122542 and A007318 (Pascal's triangle) as lower triangular matrices .

Examples

			Triangle begins :
1
1, 1
3, 4, 1
7, 13, 7, 1
17, 40, 32, 10, 1
41, 117, 124, 60, 13, 1
99, 332, 437, 286, 97, 16, 1
239, 921, 1447, 1193, 553, 143, 19, 1
577, 2512, 4584, 4556, 2682, 952, 198, 22, 1
1393, 6761, 14048, 16336, 11666, 5282, 1510, 262, 25, 1
		

Crossrefs

Cf. Columns :A001333, A119915, Diagonals : A000012, A016777, Antidiagonal sums : A077995

Formula

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(1,1) = 1 and T(n,k) = 0 if k<0 or if k>n.

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

Sum_{k, 0<=k<=n} T(n,k)*x^k = A000007(n), A001333(n), A104934(n), A122958(n), A122690(n), A091928(n) for x = -1, 0, 1, 2, 3, 4 respectively.