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Search: id:A062137
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| A062137 |
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Coefficient triangle of generalized Laguerre polynomials n!*L(n,3,x) (rising powers of x). |
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13
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| 1, 4, -1, 20, -10, 1, 120, -90, 18, -1, 840, -840, 252, -28, 1, 6720, -8400, 3360, -560, 40, -1, 60480, -90720, 45360, -10080, 1080, -54, 1, 604800, -1058400, 635040, -176400, 25200, -1890, 70, -1, 6652800, -13305600, 9313920
(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,3,x)= sum(a(n,m)*x^m,m=0..n) have e.g.f. exp(-z*x/(1-z))/(1-z)^4. 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).
These polynomials appear in the radial part of the l=1 (p-wave) eigen functions for the discrete energy levels of the H-atom. See Messiah reference.
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REFERENCES
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A. Messiah, Quantum mechanics, vol. 1, p. 419, eq.(XI.18a), North Holland, 1969.
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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+3, n-m)/m!.
E.g.f. for m-th column sequence: ((-x/(1-x))^m)/(m!*(1-x)^4), m >= 0.
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EXAMPLE
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{1}; {4,-1}; {20,-10,1}; {120,-90,18,-1}; ...; 2!*L(2,3,x)=20-10*x+x^2.
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CROSSREFS
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For m=0..5 the (unsigned) columns give A001715, A061206, A062141-A062144. The row sums (signed) give A062146, the row sums (unsigned) give A062147.
Sequence in context: A078939 A135891 A049459 this_sequence A049352 A121336 A126457
Adjacent sequences: A062134 A062135 A062136 this_sequence A062138 A062139 A062140
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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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