A096713 Irregular triangle T(n,k) of nonzero coefficients of the modified Hermite polynomials (n >= 0 and 0 <= k <= floor(n/2)).

1, 1, -1, 1, -3, 1, 3, -6, 1, 15, -10, 1, -15, 45, -15, 1, -105, 105, -21, 1, 105, -420, 210, -28, 1, 945, -1260, 378, -36, 1, -945, 4725, -3150, 630, -45, 1, -10395, 17325, -6930, 990, -55, 1, 10395, -62370, 51975, -13860, 1485, -66, 1, 135135, -270270, 135135, -25740, 2145, -78, 1

Offset: 0

Author

Eric W. Weisstein, Jul 04 2004

Keywords

• sign
• tabf

Comments

Triangle of nonzero coefficients of matching polynomial of complete graph of order n.

Row sums of absolute values produce A000085 (number of involutions). - Wouter Meeussen, Mar 12 2008

Row n has floor(n/2) + 1 nonzero coefficients. - Robert Israel, Dec 23 2015

Also the nonzero terms of the Bell matrix generated by the sequence [-1,1,0,0,0, ...] read by rows (see second Sage program). For the definition of the Bell matrix see A264428. - Peter Luschny, Jan 20 2016

From Petros Hadjicostas, Oct 28 2019: (Start)

The formulas about the p.d.f. of the standard normal distribution were proved, for example, by Charlier (1905, pp. 13-15), but they were well-known for many years before him. Charlier (1905) has generalized these results to other measures whose n-th moment (around 0) exists for each integer n >= 0.

Different forms (with or without signs) of these coefficients T(n,k) appear in other arrays as well; e.g., see A049403, A104556, A122848, A130757 (odd rows only), etc.

(End)

Examples

			Triangle T(n,k) (with rows n >= 0 and columns k >= 0) begins as follows:
    1;
    1;
   -1,     1;
   -3,     1;
    3,    -6,    1;
   15,    -10,   1;
  -15,     45, -15,   1;
  -105,   105, -21,   1;
   105,  -420, 210, -28, 1;
   945, -1260, 378, -36, 1;
   ...
The corresponding modified Hermite polynomials are as follows
He_0(x) = 1, He_1(x) = x,
He_2(x) = -1 + x^2, He_3(x) = -3*x + x^3,
He_4(x) = 3 - 6*x^2 + x^4, He_5(x) = 15*x - 10*x^3 + x^5, ...
[Modified by _Petros Hadjicostas_, Oct 28 2019]
		

References

• C. D. Godsil, Algebraic Combinatorics, Chapman & Hall, New York, 1993.

Links

• Robert Israel, Table of n, a(n) for n = 0..10099 (rows 0 to 199, flattened)
• Carl V. L. Charlier, Über die Darstellung willkürlicher Funktionen, Arkiv För Matematik, Astronomi Och Fysik, Band 2, No. 20 (Meddelande från Lunds Astronomiska Observatorium, Series I, No. 27), 1905, 1-35. [Accessible only in the USA via the HathiTrust Digital Library.]
• Nicolas Loizeau, Berislav Buča, and Dries Sels, Opening Krylov space to access all-time dynamics via dynamical symmetries, arXiv:2503.07403 [quant-ph], 2025. See p. 9.
• Tom Halverson and Theodore N. Jacobson, Set-partition tableaux and representations of diagram algebras, arXiv:1808.08118 [math.RT], 2018.
• Eric Weisstein's World of Mathematics, Hermite Polynomial.
• Eric Weisstein's World of Mathematics, Matching Polynomial. - _Eric W. Weisstein_, Sep 27 2008

Crossrefs

Cf. A000085, A049403, A104556, A122848, A130757, A264428.

Programs

Maple

A:= NULL:
for n from 0 to 20 do
  HH:= expand(orthopoly[H](n,x/sqrt(2))/2^(n/2));
  C:= subs(0=NULL, [seq(coeff(HH,x,j),j=0..n)]);
  A:= A, op(C);
od:
A; #  Robert Israel, Dec 23 2015
# Alternatively:
A096713 := (n, k) -> `if`(2*kA096713(n, k), k=0..n), n=0..13); # Peter Luschny, Dec 24 2015

Mathematica

Table[CoefficientList[HermiteH[n,x/Sqrt[2] ]/2^(n/2),x],{n,0,25}] (* Wouter Meeussen, Mar 12 2008 *)

PARI

T(n,k)=if(k<0||2*k>n, 0, (-1)^(n\2-k)*n!/(n\2-k)!/(n%2+2*k)!/2^(n\2-k)) /* Michael Somos, Jun 04 2005 */

Python

from sympy import hermite, Poly, sqrt
def a(n): return Poly(hermite(n, x/sqrt(2))/2**(n/2), x).coeffs()[::-1]
for n in range(21): print(a(n)) # Indranil Ghosh, May 26 2017

Sage

from sage.functions.hypergeometric import closed_form
def A096713_row(n):
    R. = ZZ[]
    h = hypergeometric([-n/2,(1-n)/2], [], -2*z)
    T = R(closed_form(h)).coefficients()
    return T[::-1]
for n in range(13): A096713_row(n) # Peter Luschny, Aug 21 2014

Sage

# uses[bell_transform from A264428]
def bell_zero_filter(generator, dim):
    G = [generator(k) for k in srange(dim)]
    row = lambda n: bell_transform(n, G)
    F = [filter(lambda r: r != 0, R) for R in [row(n) for n in srange(dim)]]
    return [i for f in F for i in f]
print(bell_zero_filter(lambda n: [1,-1][n] if n < 2 else 0, 14)) # Peter Luschny, Jan 20 2016

Formula

G.f.: HermiteH(n,x/sqrt(2))/2^(n/2). - Wouter Meeussen, Mar 12 2008

From Robert Israel, Dec 23 2015: (Start)

T(2*m, k) = (-1)^(m+k)*(2*m)!*2^(k-m)/((m-k)!*(2*k)!), k = 0..m.

T(2*m+1, k) = (-1)^(m+k)*(2*m+1)!*2^(k-m)/((m-k)!*(2*k+1)!), k = 0..m. (End)

From Petros Hadjicostas, Oct 28 2019: (Start)

Let He_n(x) be the n-th modified Hermite polynomial (see the references above); i.e., let He_n(x) = Sum_{k = 0..m} T(2*m, k)*x^(2*k) when n = 2*m and He_n(x) = Sum_{k = 0..m} T(2*m+1, k)*x^(2*k+1) when n = 2*m+1.

Let phi(x) = (1/sqrt(2*Pi)) * exp(-x^2/2) be the p.d.f. of a standard normal distribution. Then He_n(x) = (-1)^n * (1/phi(x)) * d^n(phi(x))/dx^n for n >= 0.

We have He_n(x) = x*He_{n-1}(x) - (n-1)*He_{n-2}(x) for n >= 2. (End)