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A129467
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Orthogonal polynomials with all zeros integers from 2*A000217.
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6
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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; internal format)
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OFFSET
| 0,5
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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
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REFERENCES
| 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.
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LINKS
| G. E. Andrews, W. Gawronski and L. L. Littlejohn, The Legendre-Stirling Numbers
W. Lang, First ten rows and more.
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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.
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EXAMPLE
| [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).
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CROSSREFS
| Row sums give A130031. Unsigned row sums give A130032.
Cf. A129462 (v=2 member), A129065 (v=1 member).
Sequence in context: A010107 A119830 A039910 * A129065 A202700 A024026
Adjacent sequences: A129464 A129465 A129466 * A129468 A129469 A129470
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KEYWORD
| sign,tabl,easy
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) May 04 2007
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