%I #72 Apr 17 2020 02:53:53

%S 1,1,-1,1,-3,1,3,-6,1,15,-10,1,-15,45,-15,1,-105,105,-21,1,105,-420,

%T 210,-28,1,945,-1260,378,-36,1,-945,4725,-3150,630,-45,1,-10395,17325,

%U -6930,990,-55,1,10395,-62370,51975,-13860,1485,-66,1,135135,-270270,135135,-25740,2145,-78,1

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

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

%C Row sums of absolute values produce A000085 (number of involutions). - _Wouter Meeussen_, Mar 12 2008

%C Row n has floor(n/2) + 1 nonzero coefficients. - _Robert Israel_, Dec 23 2015

%C 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

%C From _Petros Hadjicostas_, Oct 28 2019: (Start)

%C 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.

%C 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.

%C (End)

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

%H Robert Israel, <a href="/A096713/b096713.txt">Table of n, a(n) for n = 0..10099</a> (rows 0 to 199, flattened)

%H Carl V. L. Charlier, <a href="https://babel.hathitrust.org/cgi/pt?id=chi.73230316&amp;view=1up&amp;seq=103">Über die Darstellung willkürlicher Funktionen</a>, 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 <a href="https://www.hathitrust.org/accessibility">HathiTrust Digital Library</a>.]

%H Tom Halverson and Theodore N. Jacobson, <a href="https://arxiv.org/abs/1808.08118">Set-partition tableaux and representations of diagram algebras</a>, arXiv:1808.08118 [math.RT], 2018.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HermitePolynomial.html">Hermite Polynomial</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MatchingPolynomial.html">Matching Polynomial</a>. - _Eric W. Weisstein_, Sep 27 2008

%F G.f.: HermiteH(n,x/sqrt(2))/2^(n/2). - _Wouter Meeussen_, Mar 12 2008

%F From _Robert Israel_, Dec 23 2015: (Start)

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

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

%F From _Petros Hadjicostas_, Oct 28 2019: (Start)

%F 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.

%F 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.

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

%e Triangle T(n,k) (with rows n >= 0 and columns k >= 0) begins as follows:

%e 1;

%e 1;

%e -1, 1;

%e -3, 1;

%e 3, -6, 1;

%e 15, -10, 1;

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

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

%e 105, -420, 210, -28, 1;

%e 945, -1260, 378, -36, 1;

%e ...

%e The corresponding modified Hermite polynomials are as follows

%e He_0(x) = 1, He_1(x) = x,

%e He_2(x) = -1 + x^2, He_3(x) = -3*x + x^3,

%e He_4(x) = 3 - 6*x^2 + x^4, He_5(x) = 15*x - 10*x^3 + x^5, ...

%e [Modified by _Petros Hadjicostas_, Oct 28 2019]

%p A:= NULL:

%p for n from 0 to 20 do

%p HH:= expand(orthopoly[H](n,x/sqrt(2))/2^(n/2));

%p C:= subs(0=NULL, [seq(coeff(HH,x,j),j=0..n)]);

%p A:= A, op(C);

%p od:

%p A; # _Robert Israel_, Dec 23 2015

%p # Alternatively:

%p A096713 := (n, k) -> `if`(2*k<n, NULL, (-1/2)^(n-k)*n!/((2*k-n)!*(n-k)!)):

%p seq(seq(A096713(n, k), k=0..n), n=0..13); # _Peter Luschny_, Dec 24 2015

%t Table[CoefficientList[HermiteH[n,x/Sqrt[2] ]/2^(n/2),x],{n,0,25}] (* _Wouter Meeussen_, Mar 12 2008 *)

%o (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 */

%o (Sage)

%o from sage.functions.hypergeometric import closed_form

%o def A096713_row(n):

%o R.<z> = ZZ[]

%o h = hypergeometric([-n/2,(1-n)/2], [], -2*z)

%o T = R(closed_form(h)).coefficients()

%o return T[::-1]

%o for n in range(13): A096713_row(n) # _Peter Luschny_, Aug 21 2014

%o (Sage) # uses[bell_transform from A264428]

%o def bell_zero_filter(generator, dim):

%o G = [generator(k) for k in srange(dim)]

%o row = lambda n: bell_transform(n, G)

%o F = [filter(lambda r: r != 0, R) for R in [row(n) for n in srange(dim)]]

%o return [i for f in F for i in f]

%o print(bell_zero_filter(lambda n: [1,-1][n] if n < 2 else 0, 14)) # _Peter Luschny_, Jan 20 2016

%o (Python)

%o from sympy import hermite, Poly, sqrt

%o def a(n): return Poly(hermite(n, x/sqrt(2))/2**(n/2), x).coeffs()[::-1]

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

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

%K sign,tabf

%O 0,5

%A _Eric W. Weisstein_, Jul 04 2004