A112973 Riordan array (1/(1-x-x^2), x(1+x)/(1-x-x^2)^2).

1, 1, 1, 2, 4, 1, 3, 12, 7, 1, 5, 31, 31, 10, 1, 8, 73, 110, 59, 13, 1, 13, 162, 340, 267, 96, 16, 1, 21, 344, 956, 1022, 529, 142, 19, 1, 34, 707, 2507, 3479, 2416, 923, 197, 22, 1, 55, 1416, 6231, 10850, 9657, 4900, 1476, 261, 25, 1, 89, 2778, 14840, 31606, 34905

Offset: 0

Author

Paul Barry, Oct 07 2005

Keywords

• easy
• nonn
• tabl

Comments

Row sums are A091702. Diagonal sums are A052960. First column is A000045(n+1).

Second column is A129707. - Ralf Stephan, Dec 31 2013

Examples

			Rows begin
1;
1,1;
2,4,1;
3,12,7,1;
5,31,31,10,1;
8,73,110,59,13,1;
		

Programs

Maxima

T(n,k):=sum(binomial(m,n-m)*binomial(m+k,2*k),m,floor(n/2),n); /* Vladimir Kruchinin, Apr 21 2015 */

Formula

T(n,k) = 2*T(n-1,k) + T(n-1,k-1) + T(n-2,k) + T(n-2,k-1) - 2*T(n-3,k) - T(n-4,k), T(0,0) = T(1,0) = T(1,1) = T(2,2) = 1, T(2,0) = 2, T(2,1) = 4, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Jan 21 2014

G.f.: (x^2+x-1)/((x^2+x)*y-x^4-2*x^3+x^2+2*x-1). - Vladimir Kruchinin, Apr 21 2015

T(n,k) = Sum_{m=floor(n/2)..n} C(m,n-m)*C(m+k,2*k). - Vladimir Kruchinin, Apr 21 2015

Extensions

Definition corrected by Ralf Stephan, Dec 31 2013