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%I #10 Aug 28 2019 16:37:34
%S 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,
%T 1001,182,7,1,35,273,715,715,273,35,1,9,408,4284,15912,24310,15912,
%U 4284,408,9,5,285,3876,19380,41990,41990,19380,3876,285,5,11,770,13167,85272
%N Normalized triangle of odd numbered entries of even numbered rows of Pascal's triangle A007318.
%C b(n)= A006519(n), with b(n) defined in the formula. For every odd n b(n)=1.
%C 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.
%H W. Lang, <a href="/A091043/a091043.txt">First 9 rows</a>.
%F 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.
%e [1];[1,1];[3,10,3];[1,7,7,1];[5,60,126,60,5];...
%e 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).
%K nonn,easy,tabl
%O 1,4
%A _Wolfdieter Lang_, Jan 23 2004
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