A066325 - OEIS

%I #80 Jul 19 2024 09:17:20

%S 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 -105,0,105,0,-21,0,1,105,0,-420,0,210,0,-28,0,1,0,945,0,-1260,0,378,

%U 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 Coefficients of unitary Hermite polynomials He_n(x).

%C Also number of involutions on n labeled elements with k fixed points times (-1)^(number of 2-cycles).

%C Also called normalized Hermite polynomials.

%C 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 F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, pp. 89,94 (2.3.41,54).

%H Robert Israel, <a href="/A066325/b066325.txt">Rows n=0..140 of triangle, flattened</a>

%H 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 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 E. Elizalde, <a href="http://arXiv.org/abs/gr-qc/0409076">Cosmology: techniques and observations</a>, arXiv:gr-qc/0409076, 2004.

%H 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 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 <a href="/index/He#Hermite">Index entries for sequences related to Hermite polynomials</a>

%F T(n, k) = (-2)^((k-n)/2)*n!/(k!*((n-k)/2)!) for n-k even, 0 otherwise.

%F E.g.f. of row polynomials {He_n(y)}:  A(x, y) = exp(x*y - x^2/2).

%F 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 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 Triangle begins:

%e     1;

%e     0,     1;

%e    -1,     0,   1;

%e     0,    -3,   0,    1;

%e     3,     0,  -6,    0,   1;

%e     0,    15,   0,  -10,   0,   1;

%e   -15,     0,  45,    0, -15,   0,  1;

%e     0,  -105,   0,  105,   0, -21,  0, 1;

%e   ...

%p Q:= [seq(orthopoly[H](n,x/sqrt(2))/2^(n/2), n=0..20)]:

%p seq(seq(coeff(Q[n+1],x,k),k=0..n),n=0..20); # _Robert Israel_, Jan 01 2016

%p # Alternative:

%p T := proc(n,k) option remember; if k > n then 0 elif n = k then 1 else

%p (T(n, k+2)*(k+2)*(k+1))/(k-n) fi end:

%p seq(print(seq(T(n, k), k = 0..n)), n = 0..10); # _Peter Luschny_, Jan 08 2023

%t 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 (Sage)

%o def A066325_row(n):

%o     T = [0]*(n+1)

%o     if n==1: return [1]

%o     for m in (1..n-1):

%o         a,b,c = 1,0,0

%o         for k in range(m,-1,-1):

%o             r = a - (k+1)*c

%o             if k < m : T[k+2] = u;

%o             a,b,c = T[k-1],a,b

%o             u = r

%o         T[1] = u;

%o     return T[1:]

%o for n in (1..11): A066325_row(n)  # _Peter Luschny_, Nov 01 2012

%o (Sage) # uses[riordan_array from A256893]

%o riordan_array(exp(-x^2/2), x, 8, True) # _Peter Luschny_, Nov 23 2018

%o (Python)

%o from sympy import Poly

%o from sympy.abc import x

%o 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 def a(n): return Poly(H(n, x), x).all_coeffs()[::-1]

%o for n in range(21): print(a(n)) # _Indranil Ghosh_, May 26 2017

%o (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 Row sums: A001464 (with different signs).

%Y Row sums of absolute values: A000085.

%Y Absolute values are given in A099174.

%Y Cf. A159834, A001147, A060821 (Hermite H_n(x)).

%K sign,easy,tabl

%O 0,8

%A _Christian G. Bower_, Dec 14 2001