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A066325 Coefficients of unitary Hermite polynomials He_n(x).

%I A066325 #80 Jul 19 2024 09:17:20
%S A066325 1,0,1,-1,0,1,0,-3,0,1,3,0,-6,0,1,0,15,0,-10,0,1,-15,0,45,0,-15,0,1,0,
%T A066325 -105,0,105,0,-21,0,1,105,0,-420,0,210,0,-28,0,1,0,945,0,-1260,0,378,
%U A066325 0,-36,0,1,-945,0,4725,0,-3150,0,630,0,-45,0,1,0,-10395,0,17325,0,-6930,0,990,0,-55,0,1
%N A066325 Coefficients of unitary Hermite polynomials He_n(x).
%C A066325 Also number of involutions on n labeled elements with k fixed points times (-1)^(number of 2-cycles).
%C A066325 Also called normalized Hermite polynomials.
%C A066325 He_n(x) := H_n(x/sqrt(2)) / sqrt(2)^n, with the coefficients of H_n(x) given in A060821. See the Maple program. - _Wolfdieter Lang_, Jan 13 2020
%D A066325 F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, pp. 89,94 (2.3.41,54).
%H A066325 Robert Israel, <a href="/A066325/b066325.txt">Rows n=0..140 of triangle, flattened</a>
%H A066325 P. Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Barry1/barry97r2.html">Riordan Arrays, Orthogonal Polynomials as Moments, and Hankel Transforms</a>, J. Int. Seq. 14 (2011) # 11.2.2, chapter 8.
%H A066325 P. Diaconis and A. Gamburd, <a href="http://www.combinatorics.org/Volume_11/Abstracts/v11i2r2.html">Random matrices, magic squares and matching polynomials</a>, The Electronic Journal of Combinatorics, Volume 11, Issue 2 (2004-6), Research Paper #R2.
%H A066325 E. Elizalde, <a href="http://arXiv.org/abs/gr-qc/0409076">Cosmology: techniques and observations</a>, arXiv:gr-qc/0409076, 2004.
%H A066325 D. Foata, <a href="http://www-irma.u-strasbg.fr/~foata/paper/pub94.html">Une méthode combinatoire pour l'étude des fonctions spéciales</a>, Journées sur les méthodes en mathématiques, Institut Henri Poincaré, Paris 2-3 april 2003.
%H A066325 R. Sazdanovic, <a href="http://www.math.toronto.edu/~drorbn/SK11/Sazdanovic.pdf">A categorification of the polynomial ring</a>, slide presentation, 2011. [_Tom Copeland_, Dec 27 2015]
%H A066325 <a href="/index/He#Hermite">Index entries for sequences related to Hermite polynomials</a>
%F A066325 T(n, k) = (-2)^((k-n)/2)*n!/(k!*((n-k)/2)!) for n-k even, 0 otherwise.
%F A066325 E.g.f. of row polynomials {He_n(y)}:  A(x, y) = exp(x*y - x^2/2).
%F A066325 The umbral compositional inverses (cf. A001147) of the polynomials He(n,x) are given by the same polynomials unsigned, A099174. - _Tom Copeland_, Nov 15 2014
%F A066325 Exp(-D^2/2) x^n = He_n(x) = p_n(x+1) with D = d/dx and p_n(x), the row polynomials of A159834. These are an Appell sequence of polynomials with lowering and raising operators L = D and R = x - D. - _Tom Copeland_, Jun 26 2018
%e A066325 Triangle begins:
%e A066325     1;
%e A066325     0,     1;
%e A066325    -1,     0,   1;
%e A066325     0,    -3,   0,    1;
%e A066325     3,     0,  -6,    0,   1;
%e A066325     0,    15,   0,  -10,   0,   1;
%e A066325   -15,     0,  45,    0, -15,   0,  1;
%e A066325     0,  -105,   0,  105,   0, -21,  0, 1;
%e A066325   ...
%p A066325 Q:= [seq(orthopoly[H](n,x/sqrt(2))/2^(n/2), n=0..20)]:
%p A066325 seq(seq(coeff(Q[n+1],x,k),k=0..n),n=0..20); # _Robert Israel_, Jan 01 2016
%p A066325 # Alternative:
%p A066325 T := proc(n,k) option remember; if k > n then 0 elif n = k then 1 else
%p A066325 (T(n, k+2)*(k+2)*(k+1))/(k-n) fi end:
%p A066325 seq(print(seq(T(n, k), k = 0..n)), n = 0..10); # _Peter Luschny_, Jan 08 2023
%t A066325 H[0, x_] = 1; H[1, x_] := x; H[n_, x_] := H[n, x] = x*H[n-1, x] - (n-1)*H[n-2, x] // Expand; Table[CoefficientList[H[n, x], x], {n, 0, 11}] // Flatten (* _Jean-François Alcover_, May 11 2015 *)
%o A066325 (Sage)
%o A066325 def A066325_row(n):
%o A066325     T = [0]*(n+1)
%o A066325     if n==1: return [1]
%o A066325     for m in (1..n-1):
%o A066325         a,b,c = 1,0,0
%o A066325         for k in range(m,-1,-1):
%o A066325             r = a - (k+1)*c
%o A066325             if k < m : T[k+2] = u;
%o A066325             a,b,c = T[k-1],a,b
%o A066325             u = r
%o A066325         T[1] = u;
%o A066325     return T[1:]
%o A066325 for n in (1..11): A066325_row(n)  # _Peter Luschny_, Nov 01 2012
%o A066325 (Sage) # uses[riordan_array from A256893]
%o A066325 riordan_array(exp(-x^2/2), x, 8, True) # _Peter Luschny_, Nov 23 2018
%o A066325 (Python)
%o A066325 from sympy import Poly
%o A066325 from sympy.abc import x
%o A066325 def H(n, x): return 1 if n==0 else x if n==1 else x*H(n - 1, x) - (n - 1)*H(n - 2, x)
%o A066325 def a(n): return Poly(H(n, x), x).all_coeffs()[::-1]
%o A066325 for n in range(21): print(a(n)) # _Indranil Ghosh_, May 26 2017
%o A066325 (PARI) for(n=0, 12, for(k=0,n, print1(if(Mod(n-k,2)==0, (-2)^((k-n)/2)*n!/(k!*((n-k)/2)!), 0), ", "))) \\ _G. C. Greubel_, Nov 23 2018
%Y A066325 Row sums: A001464 (with different signs).
%Y A066325 Row sums of absolute values: A000085.
%Y A066325 Absolute values are given in A099174.
%Y A066325 Cf. A159834, A001147, A060821 (Hermite H_n(x)).
%K A066325 sign,easy,tabl
%O A066325 0,8
%A A066325 _Christian G. Bower_, Dec 14 2001