A101603 Riordan array (1/(1-x^2), x(1+x)/(1-x)).

1, 0, 1, 1, 2, 1, 0, 3, 4, 1, 1, 4, 9, 6, 1, 0, 5, 16, 19, 8, 1, 1, 6, 25, 44, 33, 10, 1, 0, 7, 36, 85, 96, 51, 12, 1, 1, 8, 49, 146, 225, 180, 73, 14, 1, 0, 9, 64, 231, 456, 501, 304, 99, 16, 1, 1, 10, 81, 344, 833, 1182, 985, 476, 129, 18, 1, 0, 11, 100, 489, 1408, 2471, 2668

Offset: 0

Author

Paul Barry, Dec 08 2004

Keywords

• easy
• nonn
• tabl

Examples

			Rows start
   1;
   0, 1;
   1, 2,  1;
   0, 3,  4,  1;
   1, 4,  9,  6, 1;
   0, 5, 16, 19, 8, 1;
		

Crossrefs

Cf. A119328 (row-reversed).

Row sums are A097076(n+1).

Diagonal sums are abs(A077902).

Programs

Mathematica

t[n_, k_] := Binomial[n+k, k]*Hypergeometric2F1[-k+1, -n, -n-k, -1]; Table[t[n-k, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 22 2013 *)

Formula

Columns are generated by x^k*(1+x)^(k-1)/(1-x)^(k+1).

T(n, k) = Sum_{j=0..n-k} C(k-1, j)*C(n-j, n-k-j).

T(n, k) = (n - k + 1)*hypergeom([1 - k, k - n], [2], 2). - Peter Luschny, Mar 09 2022