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Search: id:A062139
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| A062139 |
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Coefficient triangle of generalized Laguerre polynomials n!*L(n,2,x) (rising powers of x). |
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+0
8
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| 1, 3, -1, 12, -8, 1, 60, -60, 15, -1, 360, -480, 180, -24, 1, 2520, -4200, 2100, -420, 35, -1, 20160, -40320, 25200, -6720, 840, -48, 1, 181440, -423360, 317520, -105840, 17640, -1512, 63, -1, 1814400, -4838400, 4233600
(list; table; graph; listen)
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OFFSET
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0,2
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COMMENT
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The row polynomials s(n,x) := n!*L(n,2,x)= sum(a(n,m)*x^m,m=0..n) have e.g.f. exp(-z*x/(1-z))/(1-z)^3. They are Sheffer polynomials satisfying the binomial convolution identity s(n,x+y) = sum(binomial(n,k)*s(k,x)*p(n-k,y),k=0..n), with polynomials p(n,x)= sum(|A008297(n,m)|*(-x)^m, m=1..n), n >= 1 and p(0,x)=1 (for Sheffer polynomials see A048854 for S. Roman reference).
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LINKS
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Index entries for sequences related to Laguerre polynomials
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FORMULA
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a(n, m)=((-1)^m)*n!*binomial(n+2, n-m)/m!.
E.g.f. for m-th column sequence: ((-x/(1-x))^m)/(m!*(1-x)^3), m >= 0.
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EXAMPLE
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{1}; {3,-1}; {12,-8,1}; {60,-60,15,-1}; ...; 2!*L(2,2,x)=12-8*x+x^2.
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CROSSREFS
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For m=0..5 the (unsigned) columns give A001710, A005990, A005461, A062193-A062195. The row sums (signed) give A062197, the row sums (unsigned) give A052852.
Cf. A021009, A062137-A062140, A066667.
Sequence in context: A133366 A049458 A143492 this_sequence A156366 A144353 A039811
Adjacent sequences: A062136 A062137 A062138 this_sequence A062140 A062141 A062142
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KEYWORD
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sign,easy,tabl
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AUTHOR
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Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Jun 19 2001
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