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A137286
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Triangle of coefficients of a version of the Hermite polynomials defined by P(x, n) = x*P(x, n - 1) - n*P(x, n - 2).
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11
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1, 0, 1, -2, 0, 1, 0, -5, 0, 1, 8, 0, -9, 0, 1, 0, 33, 0, -14, 0, 1, -48, 0, 87, 0, -20, 0, 1, 0, -279, 0, 185, 0, -27, 0, 1, 384, 0, -975, 0, 345, 0, -35, 0, 1, 0, 2895, 0, -2640, 0, 588, 0, -44, 0, 1, -3840, 0, 12645, 0, -6090, 0, 938, 0, -54, 0, 1
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refs;
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history;
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internal format)
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OFFSET
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0,4
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COMMENTS
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Comments from R. J. Mathar, Jun 09 2008: (Start) Hochstadt defines the standard Hermite polynomials of A066325 via H(x,n+1)=x*H(x,n)-n*H(x,n-1); note the index shift relative to the definition in the current sequence.
As a consequence, the polynomials defined here are orthogonal with weight exp(-x^2/2) in a restricted sense than the usual Hermite Polynomials, i.e. the integral of P(x,n)*P(x,m)*exp(-x^2/2) over x=-infinity..infinity vanishes for m=n-1 (mod 2), as for any system of polynomials with separated even and odd functions, but not for the general case of m<>n as with the Hermite polynomials H(x,n) or other classical polynomials. (End)
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REFERENCES
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Harry Hochstadt, The Functions of Mathematical Physics, Dover, New York, 198, pp. 8, 42-43.
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LINKS
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Table of n, a(n) for n=0..65.
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FORMULA
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P(x,0)=1; P(x,1)=x; P(x, n) = x*P(x, n - 1) - n*P(x, n - 2)
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EXAMPLE
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{1},
{0, 1},
{-2, 0, 1},
{0, -5, 0, 1},
{8, 0, -9, 0, 1},
{0, 33, 0, -14, 0, 1},
{-48, 0, 87, 0, -20, 0, 1},
{0, -279, 0, 185, 0, -27, 0, 1},
{384, 0, -975, 0, 345, 0, -35, 0, 1},
{0, 2895, 0, -2640, 0, 588, 0, -44, 0, 1},
{-3840, 0, 12645, 0, -6090, 0, 938, 0, -54, 0, 1}
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MATHEMATICA
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P[x, 0] = 1; P[x, 1] = x; P[x_, n_] := P[x, n] = x*P[x, n - 1] - n*P[x, n - 2]; Table[ExpandAll[P[x, n]], {n, 0, 10}]; a = Table[CoefficientList[P[x, n], x], {n, 0, 10}]; Flatten[a]
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CROSSREFS
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Cf. A066325.
Sequence in context: A134317 A217377 A132277 * A180048 A128890 A196777
Adjacent sequences: A137283 A137284 A137285 * A137287 A137288 A137289
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KEYWORD
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sign,tabl,more
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AUTHOR
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Roger L. Bagula, Mar 14 2008
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EXTENSIONS
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Edited by N. J. A. Sloane, Jul 01 2008
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STATUS
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approved
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