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A066325 Coefficients of unitary Hermite polynomials He_n(x). 14
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,8
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
REFERENCES
F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, pp. 89,94 (2.3.41,54).
LINKS
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]
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
EXAMPLE
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;
...
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 *)
PROG
(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
(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
(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
CROSSREFS
Row sums: A001464 (with different signs).
Row sums of absolute values: A000085.
Absolute values are given in A099174.
Cf. A159834, A001147, A060281 (Hermite H_n(x)).
Sequence in context: A247622 A256037 A179898 * A099174 A137297 A178117
KEYWORD
sign,easy,tabl
AUTHOR
Christian G. Bower, Dec 14 2001
STATUS
approved

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Last modified May 18 21:30 EDT 2024. Contains 372666 sequences. (Running on oeis4.)