%I A137286
%S A137286 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,
%T A137286 185,0,27,0,1,384,0,975,0,345,0,35,0,1,0,2895,0,2640,0,588,0,44,0,1,
%U A137286 3840,0,12645,0,6090,0,938,0,54,0,1
%V A137286 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,
%W A137286 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,
%X A137286 938,0,-54,0,1
%N 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).
%C A137286 Comments from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 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.
%C A137286 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)
%D A137286 Harry Hochstadt, The Functions of Mathematical Physics, Dover, New York, 
               198, pp. 8, 42-43.
%F A137286 P(x,0)=1; P(x,1)=x; P(x, n) = x*P(x, n - 1) - n*P(x, n - 2)
%e A137286 {1},
%e A137286 {0, 1},
%e A137286 {-2, 0, 1},
%e A137286 {0, -5, 0, 1},
%e A137286 {8, 0, -9, 0, 1},
%e A137286 {0, 33, 0, -14, 0, 1},
%e A137286 {-48, 0, 87, 0, -20, 0, 1},
%e A137286 {0, -279, 0, 185, 0, -27, 0, 1},
%e A137286 {384, 0, -975, 0, 345, 0, -35, 0, 1},
%e A137286 {0, 2895, 0, -2640, 0, 588, 0, -44, 0, 1},
%e A137286 {-3840, 0, 12645, 0, -6090, 0, 938, 0, -54, 0, 1}
%t A137286 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]
%Y A137286 Cf. A066325.
%Y A137286 Sequence in context: A123641 A134317 A132277 this_sequence A128890 A078924 
               A137526
%Y A137286 Adjacent sequences: A137283 A137284 A137285 this_sequence A137287 A137288 
               A137289
%K A137286 sign,tabl,more
%O A137286 0,4
%A A137286 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Mar 14 2008
%E A137286 Edited by N. J. A. Sloane (njas(AT)research.att.com), Jul 01 2008