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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 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: ; [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), A191936 (row reversed?). 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 August 7 08:08 EDT 2020. Contains 336274 sequences. (Running on oeis4.)