A091043 Normalized triangle of odd numbered entries of even numbered rows of Pascal's triangle A007318.

1, 1, 1, 3, 10, 3, 1, 7, 7, 1, 5, 60, 126, 60, 5, 3, 55, 198, 198, 55, 3, 7, 182, 1001, 1716, 1001, 182, 7, 1, 35, 273, 715, 715, 273, 35, 1, 9, 408, 4284, 15912, 24310, 15912, 4284, 408, 9, 5, 285, 3876, 19380, 41990, 41990, 19380, 3876, 285, 5, 11, 770, 13167, 85272

Offset: 1

Author

Wolfdieter Lang, Jan 23 2004

Keywords

• nonn
• easy
• tabl

Comments

b(n)= A006519(n), with b(n) defined in the formula. For every odd n b(n)=1.

The row polynomials Po(n,x) := 2*b(n)*sum(a(n,m)*x^m,m=0..n-1), n>=1, appear as numerators of the generating functions for the odd numbered column sequences of array A034870. b(n) is defined in the formula below.

Examples

			[1];[1,1];[3,10,3];[1,7,7,1];[5,60,126,60,5];...
n=3: GCD(3,10,3)=GCD(3,10)=1=b(3)=A006519(3); n=4: GCD(4,28,28,4)=GCD(4,28)=4=b(4)=A006519(4).
		

Links

• W. Lang, First 9 rows.

Formula

a(n, m)= binomial(2*n, 2*m+1)/(2*b(n)), n>=m+1>=1, else 0, with b(n) := GCD(seq(binomial(2*n, 2*m+1)/2, m=0..n-1)), where GCD denotes the greatest common divisor of a set of numbers (here one half of the odd numbered entries in the even numbered rows of Pascal's triangle). It suffices to consider m=0..floor((n-1)/2) due to symmetry.