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A129467 Orthogonal polynomials with all zeros integers from 2*A000217. 12
1, 0, 1, 0, -2, 1, 0, 12, -8, 1, 0, -144, 108, -20, 1, 0, 2880, -2304, 508, -40, 1, 0, -86400, 72000, -17544, 1708, -70, 1, 0, 3628800, -3110400, 808848, -89280, 4648, -112, 1, 0, -203212800, 177811200, -48405888, 5808528, -349568, 10920, -168, 1, 0, 14631321600, -13005619200 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

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.

Apparently this is the triangle read by rows of Legendre-Stirling numbers of the first kind. See the Andrews-Gawronski-Littlejohn paper, table 2. The mirror version is the triangle A191936. - Omar E. Pol, Jan 10 2012

LINKS

Table of n, a(n) for n=0..47.

G. E. Andrews, W. Gawronski and L. L. Littlejohn, The Legendre-Stirling Numbers

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), pp. 3815-3829.

M. W. Coffey, M. C. Lettington, On Fibonacci Polynomial Expressions for Sums of mth Powers, their implications for Faulhaber's Formula and some Theorems of Fermat, arXiv:1510.05402 [math.NT], 2015.

W. Lang, First ten rows and more.

FORMULA

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.

EXAMPLE

Triangle starts:

[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).

CROSSREFS

Row sums give A130031. Unsigned row sums give A130032.

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

Sequence in context: A119830 A268435 A039910 * A129065 A202700 A024026

Adjacent sequences:  A129464 A129465 A129466 * A129468 A129469 A129470

KEYWORD

sign,tabl,easy

AUTHOR

Wolfdieter Lang, May 04 2007

STATUS

approved

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Last modified June 25 22:15 EDT 2017. Contains 288730 sequences.