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

%I A096713 #76 Mar 13 2025 12:14:01
%S A096713 1,1,-1,1,-3,1,3,-6,1,15,-10,1,-15,45,-15,1,-105,105,-21,1,105,-420,
%T A096713 210,-28,1,945,-1260,378,-36,1,-945,4725,-3150,630,-45,1,-10395,17325,
%U A096713 -6930,990,-55,1,10395,-62370,51975,-13860,1485,-66,1,135135,-270270,135135,-25740,2145,-78,1
%N A096713 Irregular triangle T(n,k) of nonzero coefficients of the modified Hermite polynomials (n >= 0 and 0 <= k <= floor(n/2)).
%C A096713 Triangle of nonzero coefficients of matching polynomial of complete graph of order n.
%C A096713 Row sums of absolute values produce A000085 (number of involutions). - _Wouter Meeussen_, Mar 12 2008
%C A096713 Row n has floor(n/2) + 1 nonzero coefficients. - _Robert Israel_, Dec 23 2015
%C A096713 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 A096713 From _Petros Hadjicostas_, Oct 28 2019: (Start)
%C A096713 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 A096713 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 A096713 (End)
%D A096713 C. D. Godsil, Algebraic Combinatorics, Chapman & Hall, New York, 1993.
%H A096713 Robert Israel, <a href="/A096713/b096713.txt">Table of n, a(n) for n = 0..10099</a> (rows 0 to 199, flattened)
%H A096713 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 A096713 Nicolas Loizeau, Berislav Buča, and Dries Sels, <a href="https://arxiv.org/abs/2503.07403">Opening Krylov space to access all-time dynamics via dynamical symmetries</a>, arXiv:2503.07403 [quant-ph], 2025. See p. 9.
%H A096713 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 A096713 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HermitePolynomial.html">Hermite Polynomial</a>.
%H A096713 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MatchingPolynomial.html">Matching Polynomial</a>. - _Eric W. Weisstein_, Sep 27 2008
%F A096713 G.f.: HermiteH(n,x/sqrt(2))/2^(n/2). - _Wouter Meeussen_, Mar 12 2008
%F A096713 From _Robert Israel_, Dec 23 2015: (Start)
%F A096713 T(2*m, k) = (-1)^(m+k)*(2*m)!*2^(k-m)/((m-k)!*(2*k)!), k = 0..m.
%F A096713 T(2*m+1, k) = (-1)^(m+k)*(2*m+1)!*2^(k-m)/((m-k)!*(2*k+1)!), k = 0..m. (End)
%F A096713 From _Petros Hadjicostas_, Oct 28 2019: (Start)
%F A096713 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 A096713 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 A096713 We have He_n(x) = x*He_{n-1}(x) - (n-1)*He_{n-2}(x) for n >= 2. (End)
%e A096713 Triangle T(n,k) (with rows n >= 0 and columns k >= 0) begins as follows:
%e A096713     1;
%e A096713     1;
%e A096713    -1,     1;
%e A096713    -3,     1;
%e A096713     3,    -6,    1;
%e A096713    15,    -10,   1;
%e A096713   -15,     45, -15,   1;
%e A096713   -105,   105, -21,   1;
%e A096713    105,  -420, 210, -28, 1;
%e A096713    945, -1260, 378, -36, 1;
%e A096713    ...
%e A096713 The corresponding modified Hermite polynomials are as follows
%e A096713 He_0(x) = 1, He_1(x) = x,
%e A096713 He_2(x) = -1 + x^2, He_3(x) = -3*x + x^3,
%e A096713 He_4(x) = 3 - 6*x^2 + x^4, He_5(x) = 15*x - 10*x^3 + x^5, ...
%e A096713 [Modified by _Petros Hadjicostas_, Oct 28 2019]
%p A096713 A:= NULL:
%p A096713 for n from 0 to 20 do
%p A096713   HH:= expand(orthopoly[H](n,x/sqrt(2))/2^(n/2));
%p A096713   C:= subs(0=NULL, [seq(coeff(HH,x,j),j=0..n)]);
%p A096713   A:= A, op(C);
%p A096713 od:
%p A096713 A; #  _Robert Israel_, Dec 23 2015
%p A096713 # Alternatively:
%p A096713 A096713 := (n, k) -> `if`(2*k<n, NULL, (-1/2)^(n-k)*n!/((2*k-n)!*(n-k)!)):
%p A096713 seq(seq(A096713(n, k), k=0..n), n=0..13); # _Peter Luschny_, Dec 24 2015
%t A096713 Table[CoefficientList[HermiteH[n,x/Sqrt[2] ]/2^(n/2),x],{n,0,25}] (* _Wouter Meeussen_, Mar 12 2008 *)
%o A096713 (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 A096713 (Sage)
%o A096713 from sage.functions.hypergeometric import closed_form
%o A096713 def A096713_row(n):
%o A096713     R.<z> = ZZ[]
%o A096713     h = hypergeometric([-n/2,(1-n)/2], [], -2*z)
%o A096713     T = R(closed_form(h)).coefficients()
%o A096713     return T[::-1]
%o A096713 for n in range(13): A096713_row(n) # _Peter Luschny_, Aug 21 2014
%o A096713 (Sage) # uses[bell_transform from A264428]
%o A096713 def bell_zero_filter(generator, dim):
%o A096713     G = [generator(k) for k in srange(dim)]
%o A096713     row = lambda n: bell_transform(n, G)
%o A096713     F = [filter(lambda r: r != 0, R) for R in [row(n) for n in srange(dim)]]
%o A096713     return [i for f in F for i in f]
%o A096713 print(bell_zero_filter(lambda n: [1,-1][n] if n < 2 else 0, 14)) # _Peter Luschny_, Jan 20 2016
%o A096713 (Python)
%o A096713 from sympy import hermite, Poly, sqrt
%o A096713 def a(n): return Poly(hermite(n, x/sqrt(2))/2**(n/2), x).coeffs()[::-1]
%o A096713 for n in range(21): print(a(n)) # _Indranil Ghosh_, May 26 2017
%Y A096713 Cf. A000085, A049403, A104556, A122848, A130757, A264428.
%K A096713 sign,tabf
%O A096713 0,5
%A A096713 _Eric W. Weisstein_, Jul 04 2004