A066325 Coefficients of unitary Hermite polynomials He_n(x).

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

Offset: 0

Author

Christian G. Bower, Dec 14 2001

Keywords

• sign
• easy
• tabl

Comments

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

Also called normalized Hermite polynomials.

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

Examples

			Triangle begins:
    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,  -105,   0,  105,   0, -21,  0, 1;
  ...
		

References

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

Links

• Robert Israel, Rows n=0..140 of triangle, flattened
• P. Barry, Riordan Arrays, Orthogonal Polynomials as Moments, and Hankel Transforms, J. Int. Seq. 14 (2011) # 11.2.2, chapter 8.
• P. Diaconis and A. Gamburd, Random matrices, magic squares and matching polynomials, The Electronic Journal of Combinatorics, Volume 11, Issue 2 (2004-6), Research Paper #R2.
• E. Elizalde, Cosmology: techniques and observations, arXiv:gr-qc/0409076, 2004.
• D. Foata, Une méthode combinatoire pour l'étude des fonctions spéciales, Journées sur les méthodes en mathématiques, Institut Henri Poincaré, Paris 2-3 april 2003.
• R. Sazdanovic, A categorification of the polynomial ring, slide presentation, 2011. [_Tom Copeland_, Dec 27 2015]
• Index entries for sequences related to Hermite polynomials

Crossrefs

Row sums: A001464 (with different signs).

Row sums of absolute values: A000085.

Absolute values are given in A099174.

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

Programs

Maple

Q:= [seq(orthopoly[H](n,x/sqrt(2))/2^(n/2), n=0..20)]:
seq(seq(coeff(Q[n+1],x,k),k=0..n),n=0..20); # Robert Israel, Jan 01 2016
# Alternative:
T := proc(n,k) option remember; if k > n then 0 elif n = k then 1 else
(T(n, k+2)*(k+2)*(k+1))/(k-n) fi end:
seq(print(seq(T(n, k), k = 0..n)), n = 0..10); # Peter Luschny, Jan 08 2023

Mathematica

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 *)

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

Python

from sympy import Poly
from sympy.abc import x
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)
def a(n): return Poly(H(n, x), x).all_coeffs()[::-1]
for n in range(21): print(a(n)) # Indranil Ghosh, May 26 2017

Sage

def A066325_row(n):
    T = [0]*(n+1)
    if n==1: return [1]
    for m in (1..n-1):
        a,b,c = 1,0,0
        for k in range(m,-1,-1):
            r = a - (k+1)*c
            if k < m : T[k+2] = u;
            a,b,c = T[k-1],a,b
            u = r
        T[1] = u;
    return T[1:]
for n in (1..11): A066325_row(n)  # Peter Luschny, Nov 01 2012

Sage

# uses[riordan_array from A256893]
riordan_array(exp(-x^2/2), x, 8, True) # Peter Luschny, Nov 23 2018

Formula

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

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

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

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