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 A137286 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). 11
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS 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) REFERENCES Harry Hochstadt, The Functions of Mathematical Physics, Dover, New York, 198, pp. 8, 42-43. LINKS FORMULA P(x,0)=1; P(x,1)=x; P(x, n) = x*P(x, n - 1) - n*P(x, n - 2) EXAMPLE {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} MATHEMATICA 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] CROSSREFS Cf. A066325. Sequence in context: A134317 A217377 A132277 * A180048 A128890 A196777 Adjacent sequences:  A137283 A137284 A137285 * A137287 A137288 A137289 KEYWORD sign,tabl,more AUTHOR Roger L. Bagula, Mar 14 2008 EXTENSIONS Edited by N. J. A. Sloane, Jul 01 2008 STATUS approved

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