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Search: id:A021012
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| A021012 |
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Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x). |
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5
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| 1, 1, -1, 2, -4, 2, 6, -18, 18, -6, 24, -96, 144, -96, 24, 120, -600, 1200, -1200, 600, -120, 720, -4320, 10800, -14400, 10800, -4320, 720, 5040, -35280, 105840, -176400, 176400, -105840, 35280, -5040, 40320, -322560, 1128960, -2257920, 2822400, -2257920, 1128960, -322560, 40320, 362880, -3265920
(list; table; graph; listen)
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OFFSET
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0,4
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COMMENT
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Triangle T(n,k), read by rows : given by [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, ...] DELTA [ -1, -1, -2, -2, -3, -3, -4, -4, -5, -5, ...], where DELTA is the operator defined in A084938 . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 14 2005
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REFERENCES
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 799.
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LINKS
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Index entries for sequences related to Laguerre polynomials
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FORMULA
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T(n, k) = (-1)^k*n!*binomial(n, k). - Vladeta Jovovic (vladeta(AT)eunet.rs), May 11 2003
Sum(k>=0); T(n, k)*T(m, k) = (n+m)! . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 14 2005
Unsigned sequence = A136572 * A007318 - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 07 2008
A136572*PS, where PS is a triangle with PS[n,k] = (-1)^k*A007318[n,k]. PS = 1/PS. [From Gerald McGarvey (gerald.mcgarvey(AT)comcast.net), Aug 20 2009]
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EXAMPLE
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1; 1,-1; 2,-4,2; 6,-18,18,-6; 24,-96,144,-96,24; ...
x^3 = 6*LaguerreL(0,x)-18*LaguerreL(1,x)+18*LaguerreL(2,x)-6*LaguerreL(3,x).
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CROSSREFS
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Columns include (essentially) A000142, A001563, A001804, A001805, A001806, A001807.
Cf. A136572.
Sequence in context: A013599 A138024 A167656 this_sequence A154120 A021416 A094756
Adjacent sequences: A021009 A021010 A021011 this_sequence A021013 A021014 A021015
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KEYWORD
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sign,tabl,easy,nice
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
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More terms from Vladeta Jovovic (vladeta(AT)eunet.rs), May 11 2003
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