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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).

%I A137286 #25 Apr 10 2020 06:18:04
%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,
%T A137286 -279,0,185,0,-27,0,1,384,0,-975,0,345,0,-35,0,1,0,2895,0,-2640,0,588,
%U A137286 0,-44,0,1,-3840,0,12645,0,-6090,0,938,0,-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 From _R. J. Mathar_, Jun 09 2008: (Start)
%C A137286 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.
%H A137286 R. Paris, <a href="http://dx.doi.org/10.1016/S0377-0427(02)00553-8">A uniform asymptotic expansion for the incomplete gamma function</a>, Journal of Computational and Applied Mathematics, 148 (2002), p. 223-239 (See p. 333 and A066325/A099174. From Tom Copeland, Jan 03 2016)
%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]
%o A137286 (PARI) polx(n) = if (n == 0, 1, if (n == 1, x, x*polx(n - 1) - n*polx(n - 2)));
%o A137286 tabl(nn) = {for (n = 0, nn, pol = polx (n); for (i = 0, n, print1(polcoeff(pol, i), ", ");); print(););} \\ _Michel Marcus_, Feb 12 2014
%o A137286 (Python)
%o A137286 from sympy import Poly
%o A137286 from sympy.abc import x
%o A137286 def P(x, n): return 1 if n==0 else x if n==1 else x*P(x, n - 1) - n*P(x, n - 2)
%o A137286 def a(n): return Poly(P(x, n), x).all_coeffs()[::-1]
%o A137286 for n in range(11): print(a(n)) # _Indranil Ghosh_, May 26 2017
%Y A137286 Cf. A066325.
%Y A137286 Cf. A099174.
%K A137286 sign,tabl
%O A137286 0,4
%A A137286 _Roger L. Bagula_, Mar 14 2008
%E A137286 Edited by _N. J. A. Sloane_, Jul 01 2008