0,5

The row polynomials p(n,x)=sum(a(n,m)*x^m,m=0..n) have the n integer zeros 2*A000217(j),j=0..n-1.

The row polynomials satisfy a three term recurrence relation which qualify them as orthogonal polynomials w.r.t. some (as yet unknown) positive measure.

Column sequences (without leading zeros) give A000007, A010790(n-1)*(-1)^(n-1), A084915(n-1)*(-1)^(n-2), A130033 for m=0..3.

M. Bruschi, F. Calogero and R. Droghei, Proof of certain Diophantine conjectures and identification of remarkable classes of orthogonal polynomials, J. Physics A, 40(2007)3815-3829.

W. Lang, First ten rows and more.

Row polynomials p(n,x):=product(x-m*(m-1),m=1..n), n>=1, p(0,x):=1.

Row polynomials p(n,x):= p(n,v=n,x) with the recurrence: p(n,v,x) = (x+2*(n-1)^2-2*(v-1)*(n-1)-v+1)*p(n-1,v,x) -((n-1)^2)*((n-1-v)^2)*p(n-2,v,x)) with p(-1,v,x)=0 and p(0,v,x)=1.

a(n,m)=[x^m] p(n,n,x), n>=m>=0, else 0.

[1]; [0,1]; [0,-2,1]; [0,12,-8,1]; [0,-144,108,-20,1]; [0,2880,-2304,508,-40,1]; ...

n=3: [0,12,-8,1]. p(3,x)=x*(12-8*x+x^2)= x*(x-2)*(x-6).

n=5: [0,2880,-2304,508,-40,1]. p(5,x)=x*(2880-2304*x+508*x^2-40*x^3+x^4)=x*(x-2)*(x-6)*(x-12)*(x-20).

Row sums give A130031. Unsigned row sums give A130032.

Cf. A129462 (v=2 member), A129065 (v=1 member).

Sequence in context: A010107 A119830 A039910 this_sequence A129065 A024026 A009829

Adjacent sequences: A129464 A129465 A129466 this_sequence A129468 A129469 A129470

sign,tabl,easy

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) May 04 2007