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Search: id:A062140
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| A062140 |
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Coefficient triangle of generalized Laguerre polynomials n!*L(n,4,x) (rising powers of x). |
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+0
13
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| 1, 5, -1, 30, -12, 1, 210, -126, 21, -1, 1680, -1344, 336, -32, 1, 15120, -15120, 5040, -720, 45, -1, 151200, -181440, 75600, -14400, 1350, -60, 1, 1663200, -2328480, 1164240, -277200, 34650, -2310, 77, -1, 19958400, -31933440
(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,4,x)= sum(a(n,m)*x^m,m=0..n) have g.f. exp(-z*x/(1-z))/(1-z)^5. 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) 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+4, n-m)/m!.
E.g.f. for m-th column sequence: ((-x/(1-x))^m)/(m!*(1-x)^5), m >= 0.
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EXAMPLE
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{1}; {5,-1}; {30,-12,1}; {210,-126,21,-1}; ...; 2!*L(2,4,x)=30-12*x+x^2.
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CROSSREFS
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For m=0..5 the (unsigned) columns give A001720(n+4), A062199, A062260-A062263. The row sums (signed) give A062265, the row sums (unsigned) give A062266.
Cf. A021009, A062137-A062139, A066667.
Sequence in context: A125906 A135892 A049460 this_sequence A049353 A027759 A066833
Adjacent sequences: A062137 A062138 A062139 this_sequence A062141 A062142 A062143
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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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